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class="page-title">Department of Mathematics</h1> <script type="text/javascript"> //<![CDATA[ var defshow = "text"; var bodycontainer = "#body"; var validhashes = ",text,undergraduatetext,graduatetext,facultystafftext,subjectstext,"; if(window.location.hash.length) { var hash = cleanHash(window.location.hash); if(validhashes.indexOf("," + hash + ",") != -1) defshow = hash; } var ss = document.styleSheets; var workss = null; for(var i=0; i < ss.length; i++) { var currmedia = ""; if (typeof ss[i].media == "string") currmedia = ss[i].media else if (typeof ss[i].media == "object" && ss[i].media.mediaText) currmedia = ss[i].media.mediaText; if((currmedia == "screen" || currmedia == "all") && (ss[i].href.indexOf("/") == 0 || ss[i].href.indexOf(document.location.host) >= 0)) { workss = ss[i]; break; } } if(workss && workss.insertRule) { workss.insertRule(".tab_content { display: none; }", workss.cssRules.length) workss.insertRule(".sectionheader { display: none; }", workss.cssRules.length) workss.insertRule("#" + defshow + "container { display: block; }", workss.cssRules.length) } else if(workss && workss.addRule) { workss.addRule(".tab_content", "display: none"); workss.addRule(".sectionheader", "display: none"); workss.addRule("#" + defshow + "container", "display: block"); } else { if (window.console) window.console.log('screen.css <link> needs attribute media="screen"'); } //]]> </script> <nav id="tabs" aria-label="Page content tabs"> <ul class="clearfix" role="tablist"> <li id="texttab" role="presentation"> <a href="#textcontainer" role="tab" aria-controls="textcontainer" onclick="return showSection('text');">Overview</a></li> <li id="undergraduatetexttab" role="presentation"> <a href="#undergraduatetextcontainer" role="tab" aria-controls="undergraduatetextcontainer" onclick="return showSection('undergraduatetext');" >Undergraduate</a></li> <li id="graduatetexttab" role="presentation"> <a href="#graduatetextcontainer" role="tab" aria-controls="graduatetextcontainer" onclick="return showSection('graduatetext');" >Graduate</a></li> <li id="facultystafftexttab" role="presentation"> <a href="#facultystafftextcontainer" role="tab" aria-controls="facultystafftextcontainer" onclick="return showSection('facultystafftext');" >Faculty/Staff</a></li> <li id="subjectstexttab" role="presentation"> <a href="#subjectstextcontainer" role="tab" aria-controls="subjectstextcontainer" onclick="return showSection('subjectstext');" >Subjects</a></li> </ul> </nav> <div id="textcontainer" class="page_content tab_content" role="tabpanel" aria-labelledby="texttab"> <p>The <a href="http://math.mit.edu/" target="_blank">Department of Mathematics</a> offers training at the undergraduate, graduate, and postgraduate levels. Its expertise covers a broad spectrum of fields ranging from the traditional areas of "pure" mathematics, such as analysis, algebra, geometry, and topology, to applied mathematics areas such as combinatorics, computational biology, fluid dynamics, theoretical computer science, and theoretical physics.</p> <p>Course 18 includes two undergraduate degrees: a Bachelor of Science in Mathematics and a Bachelor of Science in Mathematics with Computer Science. Undergraduate students may choose one of three options leading to the Bachelor of Science in Mathematics: applied mathematics, pure mathematics, or general mathematics. The general mathematics option provides a great deal of flexibility and allows students to design their own programs in conjunction with their advisors. The Mathematics with Computer Science degree is offered for students who want to pursue interests in mathematics and theoretical computer science within a single undergraduate program.</p> <p>At the graduate level, the Mathematics Department offers the PhD in Mathematics, which culminates in the exposition of original research in a dissertation. Graduate students also receive training and gain experience in the teaching of mathematics.</p> <p>The CLE Moore instructorships and Applied Mathematics instructorships bring mathematicians at the postdoctoral level to MIT and provide them with training in research and teaching.</p> </div> <div id="undergraduatetextcontainer" class="page_content tab_content" role="tabpanel" aria-labelledby="undergraduatetexttab" > <a name="undergraduatetext"></a><div class="notinpdf"> <p><a href="#mathematics-bs-course-18">Bachelor of Science in Mathematics (Course 18)</a></p> <p><a href="#mathematics-computer-science-bs-course-18-c">Bachelor of Science in Mathematics with Computer Science (Course 18-C)</a></p> <p><a href="#mathematics-minor">Minor in Mathematics</a></p> </div> <h2>Undergraduate Study</h2> <p>An undergraduate degree in mathematics provides an excellent basis for graduate work in mathematics or computer science, or for employment in such fields as finance, business, or consulting. Students' programs are arranged through consultation with their faculty advisors.</p> <p>Undergraduates in mathematics are encouraged to elect an undergraduate seminar during their junior or senior year. The experience gained from active participation in a seminar conducted by a research mathematician has proven to be valuable for students planning to pursue graduate work as well as for those going on to other careers. These seminars also provide training in the verbal and written communication of mathematics and may be used to fulfill the Communication Requirement.</p> <p>Many mathematics majors take <span class="sc_courseinline"><a href="/search/?P=18.821" title="18.821" class="bubblelink code" onclick="return showCourse(this, '18.821');">18.821</a> Project Laboratory in Mathematics</span>, which fulfills the Institute's Laboratory Requirement and counts toward the聽Communication Requirement.</p> <h3><a id="mathematics-bs-course-18" name="mathematics-bs-course-18"></a>Bachelor of Science in Mathematics (Course 18)</h3> <h4>General Mathematics Option</h4> <p>In addition to the General Institute Requirements, the requirements consist of Differential Equations, plus eight additional 12-unit subjects in Course 18 of essentially different content, including at least six advanced subjects (first decimal digit one or higher) that are distributed over at least three distinct areas (at least three distinct first decimal digits). One of these eight subjects must be Linear Algebra. This leaves available 84 units of unrestricted electives. The requirements are flexible in order to accommodate students who pursue programs that combine mathematics with a related field (such as physics, economics, or management) as well as students who are interested in both pure and applied mathematics. More details can be found on the <a href="/degree-charts/mathematics-course-18/#generalmathematicstext" target="_blank">degree chart</a>.</p> <h4>Applied Mathematics Option</h4> <p>Applied mathematics focuses on the mathematical concepts and techniques applied in science, engineering, and computer science. Particular attention is given to the following principles and their mathematical formulations: propagation, equilibrium, stability, optimization, computation, statistics, and random processes.</p> <p>Sophomores interested in applied mathematics typically enroll in <span class="sc_courseinline"><a href="/search/?P=18.200" title="18.200" class="bubblelink code" onclick="return showCourse(this, '18.200');">18.200</a> Principles of Discrete Applied Mathematics</span> and <span class="sc_courseinline"><a href="/search/?P=18.300" title="18.300" class="bubblelink code" onclick="return showCourse(this, '18.300');">18.300</a> Principles of Continuum Applied Mathematics</span>. Subject <span class="sc_courseinline"><a href="/search/?P=18.200" title="18.200" class="bubblelink code" onclick="return showCourse(this, '18.200');">18.200</a></span> is devoted to the discrete aspects of applied mathematics and may be taken concurrently with <span class="sc_courseinline"><a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> Differential Equations</span>. Subject <span class="sc_courseinline"><a href="/search/?P=18.300" title="18.300" class="bubblelink code" onclick="return showCourse(this, '18.300');">18.300</a></span>, offered in the spring term, is devoted to continuous aspects and makes considerable use of differential equations.</p> <p>The subjects in Group I of the program correspond roughly to those areas of applied mathematics that make heavy use of discrete mathematics, while Group II emphasizes those subjects that deal mainly with continuous processes. Some subjects, such as probability or numerical analysis, have both discrete and continuous aspects.</p> <p>Students planning to go on to graduate work in applied mathematics should also take some basic subjects in analysis and algebra.</p> <p>More detail on the Applied Mathematics option can be found on the <a href="/degree-charts/mathematics-course-18/#appliedmathematicstext" target="_blank">degree chart</a>.</p> <h4>Pure Mathematics Option</h4> <p>Pure (or "theoretical") mathematics is the study of the basic concepts and structure of mathematics. Its goal is to arrive at a deeper understanding and an expanded knowledge of mathematics itself.</p> <p>Traditionally, pure mathematics has been classified into three general fields: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry. The undergraduate program is designed so that students become familiar with each of these areas. Students also may wish to explore other topics such as logic, number theory, complex analysis, and subjects within applied mathematics.</p> <p>The subjects <span class="sc_courseinline"><a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> Algebra I</span> and <span class="sc_courseinline"><a href="/search/?P=18.901" title="18.901" class="bubblelink code" onclick="return showCourse(this, '18.901');">18.901</a> Introduction to Topology</span> are more advanced and should not be elected until a student has had experience with proofs, as in Real Analysis (<span class="sc_courseinline"><a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a></span>, <span class="sc_courseinline"><a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a></span>, <span class="sc_courseinline"><a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a></span> or聽<span class="sc_courseinline"><a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a></span>) or <span class="sc_courseinline"><a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a> Linear Algebra</span>.</p> <p>For more details, see the <a href="/degree-charts/mathematics-course-18/#theoreticalmathematicstext" target="_blank">degree chart</a>.</p> <h3><a id="mathematics-computer-science-bs-course-18-c" name="mathematics-computer-science-bs-course-18-c"></a>Bachelor of Science in Mathematics with Computer Science (Course 18-C)</h3> <p>Mathematics and computer science are closely related fields. Problems in computer science are often formalized and solved with mathematical methods. It is likely that many important problems currently facing computer scientists will be solved by researchers skilled in algebra, analysis, combinatorics, logic and/or probability theory, as well as computer science.</p> <p>The purpose of this program is to allow students to study a combination of these mathematical areas and potential areas of application in computer science. Required subjects include linear algebra (<span class="sc_courseinline"><a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a></span>,聽<span class="sc_courseinline"><a href="/search/?P=18.C06" title="18.C06[J]" class="bubblelink code" onclick="return showCourse(this, '18.C06');">18.C06[J]</a></span>, or <span class="sc_courseinline"><a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a></span>) because it is so broadly used, and discrete mathematics (<span class="sc_courseinline"><a href="/search/?P=18.062" title="18.062[J]" class="bubblelink code" onclick="return showCourse(this, '18.062');">18.062[J]</a></span> or <span class="sc_courseinline"><a href="/search/?P=18.200" title="18.200" class="bubblelink code" onclick="return showCourse(this, '18.200');">18.200</a></span>) to give experience with proofs and the necessary tools for analyzing algorithms. The required subjects covering complexity (<span class="sc_courseinline"><a href="/search/?P=18.404" title="18.404" class="bubblelink code" onclick="return showCourse(this, '18.404');">18.404</a> Theory of Computation</span> or <span class="sc_courseinline"><a href="/search/?P=18.400" title="18.400[J]" class="bubblelink code" onclick="return showCourse(this, '18.400');">18.400[J]</a> Computability and Complexity Theory</span>) and algorithms (<span class="sc_courseinline"><a href="/search/?P=18.410" title="18.410[J]" class="bubblelink code" onclick="return showCourse(this, '18.410');">18.410[J]</a> Design and Analysis of Algorithms</span>) provide an introduction to the most theoretical aspects of computer science.聽 We also require exposure to other areas of computer science (<span class="sc_courseinline"><a href="/search/?P=6.1020" title="6.1020" class="bubblelink code" onclick="return showCourse(this, '6.1020');">6.1020</a></span>, <span class="sc_courseinline"><a href="/search/?P=6.1800" title="6.1800" class="bubblelink code" onclick="return showCourse(this, '6.1800');">6.1800</a></span>, <span class="sc_courseinline"><a href="/search/?P=6.4100" title="6.4100" class="bubblelink code" onclick="return showCourse(this, '6.4100');">6.4100</a></span>, or <span class="sc_courseinline"><a href="/search/?P=6.3900" title="6.3900" class="bubblelink code" onclick="return showCourse(this, '6.3900');">6.3900</a></span>) where mathematical issues may also arise. More details can be found on the <a href="/degree-charts/mathematics-computer-science-course-18-c/">degree chart</a>.</p> <p>Some flexibility is allowed in this program. In particular, students may substitute the more advanced subject <span class="sc_courseinline"><a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> Algebra I</span> for <span class="sc_courseinline"><a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a> Linear Algebra</span>, and, if they already have strong theorem-proving skills, may substitute <span class="sc_courseinline"><a href="/search/?P=18.211" title="18.211" class="bubblelink code" onclick="return showCourse(this, '18.211');">18.211</a> Combinatorial Analysis</span> or <span class="sc_courseinline"><a href="/search/?P=18.212" title="18.212" class="bubblelink code" onclick="return showCourse(this, '18.212');">18.212</a> Algebraic Combinatorics</span> for <span class="sc_courseinline"><a href="/search/?P=18.062" title="18.062[J]" class="bubblelink code" onclick="return showCourse(this, '18.062');">18.062[J]</a> Mathematics for Computer Science</span> or <span class="sc_courseinline"><a href="/search/?P=18.200" title="18.200" class="bubblelink code" onclick="return showCourse(this, '18.200');">18.200</a> Principles of Discrete Applied Mathematics</span>.</p> <h3><a id="mathematics-minor" name="mathematics-minor"></a>Minor in Mathematics</h3> <p>The requirements for a Minor in Mathematics are as follows: six 12-unit subjects in mathematics, beyond the Institute's Mathematics Requirement, of essentially different content, including at least three advanced subjects (first decimal digit one or higher).</p> <p>See the Undergraduate Section for a <a href="/mit/undergraduate-education/academic-programs/minors/">general description of the minor program</a>.</p> <h3>Inquiries</h3> <p>For further information, see the <a href="http://math.mit.edu/academics/undergrad/" target="_blank">department's website</a> or contact Math Academic Services, 617-253-2416.</p> </div> <div id="graduatetextcontainer" class="page_content tab_content" role="tabpanel" aria-labelledby="graduatetexttab" > <a name="graduatetext"></a><h2>Graduate Study</h2> <p>The Mathematics Department offers programs covering a broad range of topics leading to the Doctor of Philosophy or Doctor of Science degree. Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 120 doctoral students, about two thirds are in Pure Mathematics, one third in Applied Mathematics.</p> <p>The programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, theoretical physics, and the theory of computation. In addition, many mathematically oriented subjects are offered by other departments. Students in Applied Mathematics are especially encouraged to take subjects in engineering and scientific subjects related to their research.</p> <p>All students pursue research under the supervision of the faculty and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.</p> <h3>Doctor of Philosophy or Doctor of Science</h3> <p>The requirements for the <a href="/degree-charts/phd-mathematics">Doctor or Philosophy or Doctor of Science in Mathematics</a> degree include completion of a minimum of 96 units (8 graduate subjects), an oral qualifying exam, experience in classroom teaching, a thesis proposal, and a thesis containing original research in mathematics. Additional detail about these requirements can be found on the <a href="http://math.mit.edu/academics/grad/timeline/" target="_blank">department's website</a>.</p> <p>All students pursue research under the supervision of the <a href="https://math.mit.edu/directory/faculty">faculty</a>. With the assistance of their faculty advisor, each student follows an individualized program of study encompassing the student鈥檚 area of interest. Faculty advisors may be members of the Mathematics Department or other MIT departments. Students also are encouraged to take advantage of the many <a href="https://math.mit.edu/news/seminars/">seminars and colloquia</a> at MIT and in the Boston area.</p> <p>Students typically receive their degree in five years. The first two years are spent in coursework and research, culminating in an oral examination, which must be attempted by the end of their third term and completed by the end of the second year. Following the oral examination and culmination of their research, the student forms a thesis committee. The thesis committee includes the faculty advisor and at least three other faculty members. The student defends their thesis in a public defense. The thesis must meet high professional standards and make a significant original contribution to the student鈥檚 chosen research area.</p> <p>Coursework must be completed with grades of A or B, and students are expected to maintain at least a B+ average in each semester. At most, one of the eight subjects should be a reading course. Harvard math graduate subjects may occasionally be used if taken for credit. With prior approval of the Graduate Co-chairs, one relevant advanced undergraduate math subject and relevant graduate subjects from other departments may be used. Note that subjects taken under the graduate P/D/F option cannot be used to fulfill this requirement. Students in Applied Mathematics must satisfy an <a href="https://math.mit.edu/academics/grad/timeline/plan.html">additional breadth requirement</a> as part of their plan of study.</p> <p>Teaching is an important part of the academic profession and provides excellent experience in public presentation skills. All graduate students are required to complete at least one semester of classroom teaching as part of their graduate training and are encouraged to do more.</p> <h3>Interdisciplinary Programs</h3> <h4>Computational Science and Engineering</h4> <p>Students with primary interest in computational science may also consider applying to the interdisciplinary Computational Science and Engineering (CSE) program, with which the Mathematics Department is affiliated. For more information, see the <a href="http://cse.mit.edu/programs/">CSE website</a>.</p> <h4>Mathematics and Statistics</h4> <p>The Interdisciplinary Doctoral Program in Statistics provides training in statistics, including classical statistics and probability as well as computation and data analysis, to students who wish to integrate these valuable skills into their primary academic program. The program is administered jointly by the departments of Aeronautics and Astronautics, Economics, Mathematics, Mechanical Engineering, Physics, and Political Science, and the Statistics and Data Science Center within the Institute for Data, Systems, and Society. It is open to current doctoral students in participating departments. For more information, including department-specific requirements, see the <a href="/interdisciplinary/graduate-programs/phd-statistics/">full program description</a> under Interdisciplinary Graduate Programs.</p> <h3>Financial Support</h3> <p>Financial support is guaranteed for up to five years to students making satisfactory academic progress. Financial aid after the first year is usually in the form of a teaching or research assistantship.</p> <h3>Inquiries</h3> <p>For further information, see the <a href="http://math.mit.edu/academics/grad/" target="_blank">department's website</a>聽or contact Math Academic Services, 617-253-2416.</p> </div> <div id="facultystafftextcontainer" class="page_content tab_content" role="tabpanel" aria-labelledby="facultystafftexttab" > <a name="facultystafftext"></a><div class="clearfix"><h2>Faculty and Teaching Staff</h2><div class="facitem keeptogether"><p class="noindent name">Michel X. Goemans, PhD</p><p class="hangindent title">RSA Professor of Mathematics </p><p class="hangindent title">Head, Department of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">William Minicozzi, PhD</p><p class="hangindent title">Singer Professor of Mathematics </p><p class="hangindent title">Associate Head, Department of Mathematics</p></div><h3>Professors</h3><div class="facitem keeptogether"><p class="noindent name">Martin Z. Bazant, PhD</p><p class="hangindent title">E. G. Roos Professor </p><p class="hangindent title">Professor of Chemical Engineering</p><p class="hangindent title">Professor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Bonnie Berger, PhD</p><p class="hangindent title">Simons Professor of Mathematics </p><p class="hangindent title">Member, Health Sciences and Technology Faculty</p></div><div class="facitem keeptogether"><p class="noindent name">Roman Bezrukavnikov, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Alexei Borodin, PhD</p><p class="hangindent title">Professor of Mathematics</p><p class="noindent leave">(On leave)</p></div><div class="facitem keeptogether"><p class="noindent name">John W. M. Bush, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Tobias Colding, PhD</p><p class="hangindent title">Cecil and Ida Green Distinguished Professor </p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Laurent Demanet, PhD</p><p class="hangindent title">Professor of Mathematics</p><p class="hangindent title">Professor of Earth, Atmospheric and Planetary Sciences</p></div><div class="facitem keeptogether"><p class="noindent name">J枚rn Dunkel, PhD</p><p class="hangindent title">MathWorks Professor of Mathematics </p></div><div class="facitem keeptogether"><p class="noindent name">Semyon Dyatlov, PhD</p><p class="hangindent title">Professor of Mathematics</p><p class="noindent leave">(On leave, fall)</p></div><div class="facitem keeptogether"><p class="noindent name">Alan Edelman, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Pavel I. Etingof, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Lawrence Guth, PhD</p><p class="hangindent title">Claude E. Shannon (1940) Professor of Mathematics </p></div><div class="facitem keeptogether"><p class="noindent name">Anette E. Hosoi, PhD</p><p class="hangindent title">Neil and Jane Pappalardo Professor </p><p class="hangindent title">Professor of Mechanical Engineering</p><p class="hangindent title">Professor of Mathematics</p><p class="hangindent title">Member, Institute for Data, Systems, and Society</p></div><div class="facitem keeptogether"><p class="noindent name">David S. Jerison, PhD</p><p class="hangindent title">Professor of Mathematics</p><p class="noindent leave">(On leave, spring)</p></div><div class="facitem keeptogether"><p class="noindent name">Steven G. Johnson, PhD</p><p class="hangindent title">Professor of Mathematics</p><p class="hangindent title">Professor of Physics</p></div><div class="facitem keeptogether"><p class="noindent name">Victor Kac, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Jonathan Adam Kelner, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Ju-Lee Kim, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Frank Thomson Leighton, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Aleksandr A. Logunov, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">George Lusztig, PhD</p><p class="hangindent title">Edward A. Abdun-Nur (1924) Professor of Mathematics </p></div><div class="facitem keeptogether"><p class="noindent name">Davesh Maulik, PhD</p><p class="hangindent title">Professor of Mathematics</p><p class="noindent leave">(On leave, spring)</p></div><div class="facitem keeptogether"><p class="noindent name">Haynes R. Miller, PhD</p><p class="hangindent title">Professor Post-Tenure of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Ankur Moitra, PhD</p><p class="hangindent title">Norbert Wiener Professor of Mathematics </p><p class="hangindent title">Associate Director, Institute for Data, Systems, and Society</p><p class="noindent leave">(On leave)</p></div><div class="facitem keeptogether"><p class="noindent name">Elchanan Mossel, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Tomasz S. Mrowka, PhD</p><p class="hangindent title">Professor of Mathematics</p><p class="noindent leave">(On leave)</p></div><div class="facitem keeptogether"><p class="noindent name">Pablo A. Parrilo, PhD</p><p class="hangindent title">Joseph F. and Nancy P. Keithley Professor in Electrical Engineering </p><p class="hangindent title">Professor of Electrical Engineering and Computer Science</p><p class="hangindent title">Professor of Mathematics</p><p class="hangindent title">Member, Institute for Data, Systems, and Society</p><p class="noindent leave">(On leave)</p></div><div class="facitem keeptogether"><p class="noindent name">Bjorn Poonen, PhD</p><p class="hangindent title">Distinguished Professor in Science </p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Alexander Postnikov, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Philippe Rigollet, PhD</p><p class="hangindent title">Professor of Mathematics</p><p class="hangindent title">Member, Institute for Data, Systems, and Society</p></div><div class="facitem keeptogether"><p class="noindent name">Rodolfo R. Rosales, PhD</p><p class="hangindent title">Professor Post-Tenure of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Paul Seidel, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Scott Roger Sheffield, PhD</p><p class="hangindent title">Leighton Family Professor of Mathematics </p></div><div class="facitem keeptogether"><p class="noindent name">Peter W. Shor, PhD</p><p class="hangindent title">Henry Adams Morss and Henry Adams Morss, Jr. (1934) Professor of Mathematics </p></div><div class="facitem keeptogether"><p class="noindent name">Michael Sipser, PhD</p><p class="hangindent title">Donner Professor of Mathematics </p></div><div class="facitem keeptogether"><p class="noindent name">Gigliola Staffilani, PhD</p><p class="hangindent title">Abby Rockefeller Mauz茅 Professor of Mathematics </p><p class="noindent leave">(On leave)</p></div><div class="facitem keeptogether"><p class="noindent name">Daniel W. Stroock, PhD</p><p class="hangindent title">Professor Post-Tenure of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Nike Sun, PhD</p><p class="hangindent title">Professor of Mathematics</p><p class="noindent leave">(On leave, fall)</p></div><div class="facitem keeptogether"><p class="noindent name">Martin J. Wainwright, PhD</p><p class="hangindent title">Cecil H. Green Professor in Electrical Engineering </p><p class="hangindent title">Professor of Electrical Engineering and Computer Science</p><p class="hangindent title">Professor of Mathematics</p><p class="hangindent title">Member, Institute for Data, Systems, and Society</p></div><div class="facitem keeptogether"><p class="noindent name">Zhiwei Yun, PhD</p><p class="hangindent title">Professor of Mathematics</p><p class="noindent leave">(On leave, fall)</p></div><div class="facitem keeptogether"><p class="noindent name">Wei Zhang, PhD</p><p class="hangindent title">Professor of Mathematics</p></div><h3>Associate Professors</h3><div class="facitem keeptogether"><p class="noindent name">Tristan Collins, PhD</p><p class="hangindent title">Class of 1948 Career Development Professor </p><p class="hangindent title">Associate Professor of Mathematics</p><p class="noindent leave">(On leave)</p></div><div class="facitem keeptogether"><p class="noindent name">Andrew Lawrie, PhD</p><p class="hangindent title">Associate Professor of Mathematics</p><p class="noindent leave">(On leave)</p></div><div class="facitem keeptogether"><p class="noindent name">Yufei Zhao, PhD</p><p class="hangindent title">Associate Professor of Mathematics</p><p class="noindent leave">(On leave)</p></div><h3>Assistant Professors</h3><div class="facitem keeptogether"><p class="noindent name">Shaoyun Bai, PhD</p><p class="hangindent title">Assistant Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Jacopo Borga, PhD</p><p class="hangindent title">Assistant Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Jeremy Hahn, PhD</p><p class="hangindent title">Rockwell International Career Development Assistant Professor of Mathematics </p><p class="noindent leave">(On leave, spring)</p></div><div class="facitem keeptogether"><p class="noindent name">Christoph Kehle, PhD</p><p class="hangindent title">Assistant Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Dor Minzer, PhD</p><p class="hangindent title">Assistant Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Tristan Ozuch-Meersseman, PhD</p><p class="hangindent title">Assistant Professor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">John Urschel, PhD</p><p class="hangindent title">Assistant Professor of Mathematics</p><p class="noindent leave">(On leave)</p></div><h3>Visiting Professors</h3><div class="facitem keeptogether"><p class="noindent name">Yakov Varshavsky, PhD</p><p class="hangindent title">Visiting Simons Professor of Mathematics </p></div><h3>Adjunct Professors</h3><div class="facitem keeptogether"><p class="noindent name">Henry Cohn, PhD</p><p class="hangindent title">Adjunct Professor of Mathematics</p></div><h3>Lecturers</h3><div class="facitem keeptogether"><p class="noindent name">Jonathan Bloom, PhD</p><p class="hangindent title">Lecturer in Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Slava Gerovitch, PhD</p><p class="hangindent title">Lecturer in Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Peter J. Kempthorne, PhD</p><p class="hangindent title">Lecturer in Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Tanya Khovanova, PhD</p><p class="hangindent title">Lecturer in Mathematics</p></div><h3>CLE Moore Instructors</h3><div class="facitem keeptogether"><p class="noindent name">Sky Yang Cao, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Marjorie Drake, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Wenkui Du, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Giada Franz, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Felipe Hernandez, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Ruojing Jiang, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Konstantinos Kavvadias, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Zhengjiang Lin, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Thomas Massoni, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Miguel Moreira, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Jia Shi, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Jingze Zhu, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Jonathan Zung, PhD</p><p class="hangindent title">CLE Moore Instructor of Mathematics</p></div><h3>Instructors</h3><div class="facitem keeptogether"><p class="noindent name">Lior Alon, PhD</p><p class="hangindent title">Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Karol Bacik, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Mitali Bafna, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Elijah Bodish, PhD</p><p class="hangindent title">Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Pengning Chao, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Ting-Wei Chao, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Shi Chen, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Ziang Chen, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Nicholas Derr, PhD</p><p class="hangindent title">Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Manik Dhar, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Ziqi Fang, PhD</p><p class="hangindent title">Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Ludovico Giorgini, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Mohammad Reza Karimi Jaghargh, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Anya Katsevich, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Max Lipton, PhD</p><p class="hangindent title">Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Sammy Luo, PhD</p><p class="hangindent title">Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Ryan Maguire, PhD</p><p class="hangindent title">Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Shivam Nadimpalli, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Alex Pieloch, PhD</p><p class="hangindent title">Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Bauyrzhan Primkulov, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Arpon Raksit, PhD</p><p class="hangindent title">Instructor of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Michael Simkin, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Foster Tom, PhD</p><p class="hangindent title">Instructor of Applied Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Robin Zhang, PhD</p><p class="hangindent title">Instructor of Mathematics</p></div><h2>Research Staff</h2><h3>Principal Research Scientists</h3><div class="facitem keeptogether"><p class="noindent name">Andrew Victor Sutherland II, PhD</p><p class="hangindent title">Principal Research Scientist of Mathematics</p></div><h3>Research Scientists</h3><div class="facitem keeptogether"><p class="noindent name">Eran Assaf, PhD</p><p class="hangindent title">Research Scientist of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Keaton Burns, PhD</p><p class="hangindent title">Research Scientist of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Edgar Costa, PhD</p><p class="hangindent title">Research Scientist of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">David Roe, PhD</p><p class="hangindent title">Research Scientist of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Samuel Schiavone, PhD</p><p class="hangindent title">Research Scientist of Mathematics</p></div><h2>Professors Emeriti</h2><div class="facitem keeptogether"><p class="noindent name">Michael Artin, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Hung Cheng, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Daniel Z. Freedman, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p><p class="hangindent title">Professor Emeritus of Physics</p></div><div class="facitem keeptogether"><p class="noindent name">Harvey P. Greenspan, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Victor W. Guillemin, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Steven L. Kleiman, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Daniel J. Kleitman, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Richard B. Melrose, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">James R. Munkres, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Richard P. Stanley, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Harold Stark, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Gilbert Strang, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">Alar Toomre, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div><div class="facitem keeptogether"><p class="noindent name">David A. Vogan, PhD</p><p class="hangindent title">Professor Emeritus of Mathematics</p></div></div> </div> <div id="subjectstextcontainer" class="page_content tab_content" role="tabpanel" aria-labelledby="subjectstexttab" > <a name="subjectstext"></a><div id="sc_sccoursedescs"> <h2 class="sectionhead">General Mathematics</h2><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.01 Calculus</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">5-0-7 units. CALC I</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.01A" title="18.01A" class="bubblelink code" onclick="return showCourse(this, '18.01A');">18.01A</a>, <a href="/search/?P=CC.1801" title="CC.1801" class="bubblelink code" onclick="return showCourse(this, 'CC.1801');">CC.1801</a>, <a href="/search/?P=ES.1801" title="ES.1801" class="bubblelink code" onclick="return showCourse(this, 'ES.1801');">ES.1801</a>, <a href="/search/?P=ES.181A" title="ES.181A" class="bubblelink code" onclick="return showCourse(this, 'ES.181A');">ES.181A</a></span> </p> <p class="courseblockdesc">Differentiation and integration of functions of one variable, with applications. Informal treatment of limits and continuity. Differentiation: definition, rules, application to graphing, rates, approximations, and extremum problems. Indefinite integration; separable first-order differential equations. Definite integral; fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Polar coordinates. L'Hopital's rule. Improper integrals. Infinite series: geometric, p-harmonic, simple comparison tests, power series for some elementary functions. </p> <p class="courseblockinstructors seemore"><span><i>Fall: L. Guth. Spring: Information: W. Minicozzi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.01A Calculus</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Knowledge of differentiation and elementary integration </span><br/><span class="courseblockterms">U (Fall; first half of term)</span><br/><span class="courseblockhours">5-0-7 units. CALC I</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.01" title="18.01" class="bubblelink code" onclick="return showCourse(this, '18.01');">18.01</a>, <a href="/search/?P=CC.1801" title="CC.1801" class="bubblelink code" onclick="return showCourse(this, 'CC.1801');">CC.1801</a>, <a href="/search/?P=ES.1801" title="ES.1801" class="bubblelink code" onclick="return showCourse(this, 'ES.1801');">ES.1801</a>, <a href="/search/?P=ES.181A" title="ES.181A" class="bubblelink code" onclick="return showCourse(this, 'ES.181A');">ES.181A</a></span> </p> <p class="courseblockdesc">Six-week review of one-variable calculus, emphasizing material not on the high-school AB syllabus: integration techniques and applications, improper integrals, infinite series, applications to other topics, such as probability and statistics, as time permits. Prerequisites: one year of high-school calculus or the equivalent, with a score of 5 on the AB Calculus test (or the AB portion of the BC test, or an equivalent score on a standard international exam), or equivalent college transfer credit, or a passing grade on the first half of the <a href="/search/?P=18.01" title="18.01" class="bubblelink code" onclick="return showCourse(this, '18.01');">18.01</a> advanced standing exam. </p> <p class="courseblockinstructors seemore"><span><i>A. Moitra</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.02 Calculus</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.01|18.01A|18.014" onclick="return showCourse(this, '18.01|18.01A|18.014');">Calculus I (GIR)</a> </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">5-0-7 units. CALC II</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.022" title="18.022" class="bubblelink code" onclick="return showCourse(this, '18.022');">18.022</a>, <a href="/search/?P=18.02A" title="18.02A" class="bubblelink code" onclick="return showCourse(this, '18.02A');">18.02A</a>, <a href="/search/?P=CC.1802" title="CC.1802" class="bubblelink code" onclick="return showCourse(this, 'CC.1802');">CC.1802</a>, <a href="/search/?P=ES.1802" title="ES.1802" class="bubblelink code" onclick="return showCourse(this, 'ES.1802');">ES.1802</a>, <a href="/search/?P=ES.182A" title="ES.182A" class="bubblelink code" onclick="return showCourse(this, 'ES.182A');">ES.182A</a></span> </p> <p class="courseblockdesc">Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications. </p> <p class="courseblockinstructors seemore"><span><i>Fall: S Dyatlov. Spring: D Jerison</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.02A Calculus</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.01|18.01A|18.014" onclick="return showCourse(this, '18.01|18.01A|18.014');">Calculus I (GIR)</a> </span><br/><span class="courseblockterms">U (Fall, IAP, Spring; second half of term)</span><br/><span class="courseblockhours">5-0-7 units. CALC II</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.02" title="18.02" class="bubblelink code" onclick="return showCourse(this, '18.02');">18.02</a>, <a href="/search/?P=18.022" title="18.022" class="bubblelink code" onclick="return showCourse(this, '18.022');">18.022</a>, <a href="/search/?P=CC.1802" title="CC.1802" class="bubblelink code" onclick="return showCourse(this, 'CC.1802');">CC.1802</a>, <a href="/search/?P=ES.1802" title="ES.1802" class="bubblelink code" onclick="return showCourse(this, 'ES.1802');">ES.1802</a>, <a href="/search/?P=ES.182A" title="ES.182A" class="bubblelink code" onclick="return showCourse(this, 'ES.182A');">ES.182A</a></span> </p> <p class="courseblockdesc">First half is taught during the last six weeks of the Fall term; covers material in the first half of <a href="/search/?P=18.02" title="18.02" class="bubblelink code" onclick="return showCourse(this, '18.02');">18.02</a> (through double integrals). Second half of <a href="/search/?P=18.02A" title="18.02A" class="bubblelink code" onclick="return showCourse(this, '18.02A');">18.02A</a> can be taken either during IAP (daily lectures) or during the second half of the Spring term; it covers the remaining material in <a href="/search/?P=18.02" title="18.02" class="bubblelink code" onclick="return showCourse(this, '18.02');">18.02</a>. </p> <p class="courseblockinstructors seemore"><span><i>Fall, IAP: J. W. M. Bush. Spring: D. Jerison</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.022 Calculus</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.01|18.01A|18.014" onclick="return showCourse(this, '18.01|18.01A|18.014');">Calculus I (GIR)</a> </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">5-0-7 units. CALC II</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.02" title="18.02" class="bubblelink code" onclick="return showCourse(this, '18.02');">18.02</a>, <a href="/search/?P=18.02A" title="18.02A" class="bubblelink code" onclick="return showCourse(this, '18.02A');">18.02A</a>, <a href="/search/?P=CC.1802" title="CC.1802" class="bubblelink code" onclick="return showCourse(this, 'CC.1802');">CC.1802</a>, <a href="/search/?P=ES.1802" title="ES.1802" class="bubblelink code" onclick="return showCourse(this, 'ES.1802');">ES.1802</a>, <a href="/search/?P=ES.182A" title="ES.182A" class="bubblelink code" onclick="return showCourse(this, 'ES.182A');">ES.182A</a></span> </p> <p class="courseblockdesc">Calculus of several variables. Topics as in <a href="/search/?P=18.02" title="18.02" class="bubblelink code" onclick="return showCourse(this, '18.02');">18.02</a> but with more focus on mathematical concepts. Vector algebra, dot product, matrices, determinant. Functions of several variables, continuity, differentiability, derivative. Parametrized curves, arc length, curvature, torsion. Vector fields, gradient, curl, divergence. Multiple integrals, change of variables, line integrals, surface integrals. Stokes' theorem in one, two, and three dimensions. </p> <p class="courseblockinstructors seemore"><span><i>W. Minicozzi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.03 Differential Equations</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None. <i>Coreq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a></i> </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">5-0-7 units. REST</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=CC.1803" title="CC.1803" class="bubblelink code" onclick="return showCourse(this, 'CC.1803');">CC.1803</a>, <a href="/search/?P=ES.1803" title="ES.1803" class="bubblelink code" onclick="return showCourse(this, 'ES.1803');">ES.1803</a></span> </p> <p class="courseblockdesc">Study of differential equations, including modeling physical systems. Solution of first-order ODEs by analytical, graphical, and numerical methods. Linear ODEs with constant coefficients. Complex numbers and exponentials. Inhomogeneous equations: polynomial, sinusoidal, and exponential inputs. Oscillations, damping, resonance. Fourier series. Matrices, eigenvalues, eigenvectors, diagonalization. First order linear systems: normal modes, matrix exponentials, variation of parameters. Heat equation, wave equation. Nonlinear autonomous systems: critical point analysis, phase plane diagrams. </p> <p class="courseblockinstructors seemore"><span><i>Fall: J. Dunkel. Spring: L. Demanet</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.031 System Functions and the Laplace Transform</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None. <i>Coreq: <a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a></i> </span><br/><span class="courseblockterms">U (IAP)</span><br/><span class="courseblockhours">1-0-2 units</span> </p> <p class="courseblockdesc">Studies basic continuous control theory as well as representation of functions in the complex frequency domain. Covers generalized functions, unit impulse response, and convolution; and Laplace transform, system (or transfer) function, and the pole diagram. Includes examples from mechanical and electrical engineering. </p> <p class="courseblockinstructors seemore"><span><i>Information: H. R. Miller</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.032 Differential Equations</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None. <i>Coreq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a></i> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">5-0-7 units. REST</span> </p> <p class="courseblockdesc">Covers much of the same material as <a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> with more emphasis on theory. The point of view is rigorous and results are proven. Local existence and uniqueness of solutions. </p> <p class="courseblockinstructors seemore"><span><i>A. Lawrie</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.04 Complex Variables with Applications</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">4-0-8 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.075" title="18.075" class="bubblelink code" onclick="return showCourse(this, '18.075');">18.075</a>, <a href="/search/?P=18.0751" title="18.0751" class="bubblelink code" onclick="return showCourse(this, '18.0751');">18.0751</a></span> </p> <p class="courseblockdesc">Complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis, Laplace transforms, and partial differential equations. </p> <p class="courseblockinstructors seemore"><span><i>H. Cheng</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.05 Introduction to Probability and Statistics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">4-0-8 units. REST</span> </p> <p class="courseblockdesc">Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression. </p> <p class="courseblockinstructors seemore"><span><i>J. French</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.06 Linear Algebra</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">4-0-8 units. REST</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=6.C06" title="6.C06[J]" class="bubblelink code" onclick="return showCourse(this, '6.C06');">6.C06[J]</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, <a href="/search/?P=18.C06" title="18.C06[J]" class="bubblelink code" onclick="return showCourse(this, '18.C06');">18.C06[J]</a>, <a href="/search/?P=ES.1806" title="ES.1806" class="bubblelink code" onclick="return showCourse(this, 'ES.1806');">ES.1806</a></span> </p> <p class="courseblockdesc">Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, singular value decomposition, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses linear algebra software. Compared with <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, more emphasis on matrix algorithms and many applications. </p> <p class="courseblockinstructors seemore"><span><i>Fall: TBD. Spring: A. Borodin</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.C06[J] Linear Algebra and Optimization</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.C06J" title="6.C06[J]" class="bubblelink code" onclick="return showCourse(this, '6.C06J');">6.C06[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">5-0-7 units. REST</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, <a href="/search/?P=ES.1806" title="ES.1806" class="bubblelink code" onclick="return showCourse(this, 'ES.1806');">ES.1806</a></span> </p> <p class="courseblockdesc">Introductory course in linear algebra and optimization, assuming no prior exposure to linear algebra and starting from the basics, including vectors, matrices, eigenvalues, singular values, and least squares. Covers the basics in optimization including convex optimization, linear/quadratic programming, gradient descent, and regularization, building on insights from linear algebra. Explores a variety of applications in science and engineering, where the tools developed give powerful ways to understand complex systems and also extract structure from data. </p> <p class="courseblockinstructors seemore"><span><i>A. Moitra, P. Parrilo</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.062[J] Mathematics for Computer Science</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.1200J" title="6.1200[J]" class="bubblelink code" onclick="return showCourse(this, '6.1200J');">6.1200[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.01|18.01A|18.014" onclick="return showCourse(this, '18.01|18.01A|18.014');">Calculus I (GIR)</a> </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">5-0-7 units. REST</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=6.1200J" title="6.1200[J]" class="bubblelink code" onclick="return showCourse(this, '6.1200J');">6.1200[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>Z. R.聽Abel,聽F. T. Leighton,聽A. Moitra</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.063 Matrix Calculus for Machine Learning and Beyond (New)</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a> </span><br/><span class="courseblockterms">U (IAP; partial term)</span><br/><span class="courseblockhours">1-0-2 units</span> </p> <p class="courseblockdesc">Covers a coherent approach to matrix calculus, showing techniques that allow the student to think of a matrix holistically, rather than as an array of scalars; generalize and compute derivatives of important matrix factorizations and other complicated-looking operations; and understand how differentiation formulas must be reimagined in large-scale computing. Discusses "adjoint" or "reverse-mode" differentiation, custom vector-Jacobian products, and how modern automatic differentiation is more computer science than calculus. </p> <p class="courseblockinstructors seemore"><span><i>S. Johnson, A. Edelman</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.0651" title="18.0651" class="bubblelink code" onclick="return showCourse(this, '18.0651');">18.0651</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Reviews linear algebra with applications to life sciences, finance, engineering, and big data. Covers singular value decomposition, weighted least squares, signal and image processing, principal component analysis, covariance and correlation matrices, directed and undirected graphs, matrix factorizations, neural nets, machine learning, and computations with large matrices. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.0651 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.065" title="18.065" class="bubblelink code" onclick="return showCourse(this, '18.065');">18.065</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Reviews linear algebra with applications to life sciences, finance, engineering, and big data. Covers singular value decomposition, weighted least squares, signal and image processing, principal component analysis, covariance and correlation matrices, directed and undirected graphs, matrix factorizations, neural nets, machine learning, and computations with large matrices. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.065" title="18.065" class="bubblelink code" onclick="return showCourse(this, '18.065');">18.065</a>. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.075 Methods for Scientists and Engineers</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.0751" title="18.0751" class="bubblelink code" onclick="return showCourse(this, '18.0751');">18.0751</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and <a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.04" title="18.04" class="bubblelink code" onclick="return showCourse(this, '18.04');">18.04</a></span> </p> <p class="courseblockdesc">Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations. </p> <p class="courseblockinstructors seemore"><span><i>H. Cheng</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.0751 Methods for Scientists and Engineers</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.075" title="18.075" class="bubblelink code" onclick="return showCourse(this, '18.075');">18.075</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and <a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.04" title="18.04" class="bubblelink code" onclick="return showCourse(this, '18.04');">18.04</a></span> </p> <p class="courseblockdesc">Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations. Students in Courses 6, 8, 12, 18, and 22 must register for undergraduate version, <a href="/search/?P=18.075" title="18.075" class="bubblelink code" onclick="return showCourse(this, '18.075');">18.075</a>. </p> <p class="courseblockinstructors seemore"><span><i>H. Cheng</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.085 Computational Science and Engineering I</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.0851" title="18.0851" class="bubblelink code" onclick="return showCourse(this, '18.0851');">18.0851</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) </span><br/><span class="courseblockterms">U (Fall, Spring, Summer)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications. </p> <p class="courseblockinstructors seemore"><span><i>Fall: D. Kouskoulas.聽Spring: Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.0851 Computational Science and Engineering I</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.085" title="18.085" class="bubblelink code" onclick="return showCourse(this, '18.085');">18.085</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) </span><br/><span class="courseblockterms">G (Fall, Spring, Summer)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.085" title="18.085" class="bubblelink code" onclick="return showCourse(this, '18.085');">18.085</a>. </p> <p class="courseblockinstructors seemore"><span><i>Fall: D. Kouskoulas. Spring: Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.086 Computational Science and Engineering II</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.0861" title="18.0861" class="bubblelink code" onclick="return showCourse(this, '18.0861');">18.0861</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) </span><br/><span class="courseblockterms">U (Spring)<br/> Not offered regularly; consult department</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods. </p> <p class="courseblockinstructors seemore"><span><i>Information: W. G. Strang</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.0861 Computational Science and Engineering II</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.086" title="18.086" class="bubblelink code" onclick="return showCourse(this, '18.086');">18.086</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) </span><br/><span class="courseblockterms">G (Spring)<br/> Not offered regularly; consult department</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.086" title="18.086" class="bubblelink code" onclick="return showCourse(this, '18.086');">18.086</a>. </p> <p class="courseblockinstructors seemore"><span><i>Information: W. G. Strang</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.089 Review of Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Summer)</span><br/><span class="courseblockhours">5-0-7 units</span> </p> <p class="courseblockdesc">One-week review of one-variable calculus (<a href="/search/?P=18.01" title="18.01" class="bubblelink code" onclick="return showCourse(this, '18.01');">18.01</a>), followed by concentrated study covering multivariable calculus (<a href="/search/?P=18.02" title="18.02" class="bubblelink code" onclick="return showCourse(this, '18.02');">18.02</a>), two hours per day for five weeks. Primarily for graduate students in Course 2N. Degree credit allowed only in special circumstances. </p> <p class="courseblockinstructors seemore"><span><i>Information: W. Minicozzi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.090 Introduction to Mathematical Reasoning</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None. <i>Coreq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a></i> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units. REST</span> </p> <p class="courseblockdesc">Focuses on understanding and constructing mathematical arguments. Discusses foundational topics (such as infinite sets, quantifiers, and methods of proof) as well as selected concepts from algebra (permutations, vector spaces, fields) and analysis (sequences of real numbers). Particularly suitable for students desiring additional experience with proofs before going on to more advanced mathematics subjects or subjects in related areas with significant mathematical content. </p> <p class="courseblockinstructors seemore"><span><i>S. Dyatlov, B. Poonen, P. Seidel</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.091 Introduction to Metric Spaces (New)</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a> </span><br/><span class="courseblockterms">U (IAP; partial term)</span><br/><span class="courseblockhours">1-0-2 units</span> </p> <p class="courseblockdesc">Covers metrics, open and closed sets, continuous functions (from a topological perspective), function spaces, completeness, and compactness. Aims to provide more complex concepts and proofs for students who have taken <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a> as their real analysis subject. </p> <p class="courseblockinstructors seemore"><span><i>W. Minicozzi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.094[J] Teaching College-Level Science and Engineering</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=1.95J" title="1.95[J]" class="bubblelink code" onclick="return showCourse(this, '1.95J');">1.95[J]</a>, <a href="/search/?P=5.95J" title="5.95[J]" class="bubblelink code" onclick="return showCourse(this, '5.95J');">5.95[J]</a>, <a href="/search/?P=7.59J" title="7.59[J]" class="bubblelink code" onclick="return showCourse(this, '7.59J');">7.59[J]</a>, <a href="/search/?P=8.395J" title="8.395[J]" class="bubblelink code" onclick="return showCourse(this, '8.395J');">8.395[J]</a><br/> Subject meets with <a href="/search/?P=2.978" title="2.978" class="bubblelink code" onclick="return showCourse(this, '2.978');">2.978</a></span><br/><span class="courseblockprereq">Prereq: None </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">2-0-2 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=5.95J" title="5.95[J]" class="bubblelink code" onclick="return showCourse(this, '5.95J');">5.95[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>J. Rankin</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.095 Mathematics Lecture Series</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.01|18.01A|18.014" onclick="return showCourse(this, '18.01|18.01A|18.014');">Calculus I (GIR)</a> </span><br/><span class="courseblockterms">U (IAP)</span><br/><span class="courseblockhours">2-0-4 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Ten lectures by mathematics faculty members on interesting topics from both classical and modern mathematics. All lectures accessible to students with calculus background and an interest in mathematics. At each lecture, reading and exercises are assigned. Students prepare these for discussion in a weekly problem session. </p> <p class="courseblockinstructors seemore"><span><i>Information: W. Minicozzi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.098 Internship in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">U (Fall, IAP, Spring, Summer)</span><br/><span class="courseblockhours">Units arranged [P/D/F]</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Provides academic credit for students pursuing internships to gain practical experience in the applications of mathematical concepts and methods. </p> <p class="courseblockinstructors seemore"><span><i>Information: W. Minicozzi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.099 Independent Study</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">U (Fall, IAP, Spring, Summer)</span><br/><span class="courseblockhours">Units arranged</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Studies (during IAP) or special individual reading (during regular terms). Arranged in consultation with individual faculty members and subject to departmental approval.聽 May not be used to satisfy Mathematics major requirements. </p> <p class="courseblockinstructors seemore"><span><i>Information: W. Minicozzi</i></span></p> </div><h2 class="sectionhead">Analysis</h2><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.1001 Real Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">G (Fall, Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.1002" title="18.1002" class="bubblelink code" onclick="return showCourse(this, '18.1002');">18.1002</a>, <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a></span> </p> <p class="courseblockdesc">Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>. Gives applications where possible. Concerned primarily with the real line. Students in Course 18 must register for undergraduate version <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>. </p> <p class="courseblockinstructors seemore"><span><i>Fall: Q. Deng. Spring: J. Zhu</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.1002 Real Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">G (Fall, Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.1001" title="18.1001" class="bubblelink code" onclick="return showCourse(this, '18.1001');">18.1001</a>, <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a></span> </p> <p class="courseblockdesc">Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Students in Course 18 must register for undergraduate version <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>. </p> <p class="courseblockinstructors seemore"><span><i>Fall: R. Melrose. Spring: G. Franz</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.100A Real Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.1001" title="18.1001" class="bubblelink code" onclick="return showCourse(this, '18.1001');">18.1001</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.1001" title="18.1001" class="bubblelink code" onclick="return showCourse(this, '18.1001');">18.1001</a>, <a href="/search/?P=18.1002" title="18.1002" class="bubblelink code" onclick="return showCourse(this, '18.1002');">18.1002</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a></span> </p> <p class="courseblockdesc">Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>. Gives applications where possible. Concerned primarily with the real line. </p> <p class="courseblockinstructors seemore"><span><i>Fall: Q. Deng. Spring: J. Zhu</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.100B Real Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.1002" title="18.1002" class="bubblelink code" onclick="return showCourse(this, '18.1002');">18.1002</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.1001" title="18.1001" class="bubblelink code" onclick="return showCourse(this, '18.1001');">18.1001</a>, <a href="/search/?P=18.1002" title="18.1002" class="bubblelink code" onclick="return showCourse(this, '18.1002');">18.1002</a>, <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a></span> </p> <p class="courseblockdesc">Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. </p> <p class="courseblockinstructors seemore"><span><i>Fall: R. Melrose. Spring: G. Franz</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.100P Real Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">4-0-11 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.1001" title="18.1001" class="bubblelink code" onclick="return showCourse(this, '18.1001');">18.1001</a>, <a href="/search/?P=18.1002" title="18.1002" class="bubblelink code" onclick="return showCourse(this, '18.1002');">18.1002</a>, <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a></span> </p> <p class="courseblockdesc">Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>. Gives applications where possible. Concerned primarily with the real line. Includes instruction and practice in written communication. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>K. Naff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.100Q Real Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">4-0-11 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.1001" title="18.1001" class="bubblelink code" onclick="return showCourse(this, '18.1001');">18.1001</a>, <a href="/search/?P=18.1002" title="18.1002" class="bubblelink code" onclick="return showCourse(this, '18.1002');">18.1002</a>, <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a></span> </p> <p class="courseblockdesc">Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Includes instruction and practice in written communication. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>C. Oh</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.101 Analysis and Manifolds</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.1011" title="18.1011" class="bubblelink code" onclick="return showCourse(this, '18.1011');">18.1011</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to the theory of manifolds: vector fields and densities on manifolds, integral calculus in the manifold setting and the manifold version of the divergence theorem. <a href="/search/?P=18.901" title="18.901" class="bubblelink code" onclick="return showCourse(this, '18.901');">18.901</a> helpful but not required. </p> <p class="courseblockinstructors seemore"><span><i>M. Jezequel</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.1011 Analysis and Manifolds</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.101" title="18.101" class="bubblelink code" onclick="return showCourse(this, '18.101');">18.101</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to the theory of manifolds: vector fields and densities on manifolds, integral calculus in the manifold setting and the manifold version of the divergence theorem. <a href="/search/?P=18.9011" title="18.9011" class="bubblelink code" onclick="return showCourse(this, '18.9011');">18.9011</a> helpful but not required. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.101" title="18.101" class="bubblelink code" onclick="return showCourse(this, '18.101');">18.101</a>. </p> <p class="courseblockinstructors seemore"><span><i>M. Jezequel</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.102 Introduction to Functional Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.1021" title="18.1021" class="bubblelink code" onclick="return showCourse(this, '18.1021');">18.1021</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators. Lebesgue measure, measurable functions, integrability, completeness of L-p spaces. Hilbert space. Compact, Hilbert-Schmidt and trace class operators. Spectral theorem. </p> <p class="courseblockinstructors seemore"><span><i>M. Jezequel</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.1021 Introduction to Functional Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.102" title="18.102" class="bubblelink code" onclick="return showCourse(this, '18.102');">18.102</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators. Lebesgue measure, measurable functions, integrability, completeness of L-p spaces. Hilbert space. Compact, Hilbert-Schmidt and trace class operators. Spectral theorem. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.102" title="18.102" class="bubblelink code" onclick="return showCourse(this, '18.102');">18.102</a>. </p> <p class="courseblockinstructors seemore"><span><i>M. Jezequel</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.103 Fourier Analysis: Theory and Applications</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.1031" title="18.1031" class="bubblelink code" onclick="return showCourse(this, '18.1031');">18.1031</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals. </p> <p class="courseblockinstructors seemore"><span><i>J. Shi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.1031 Fourier Analysis: Theory and Applications</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.103" title="18.103" class="bubblelink code" onclick="return showCourse(this, '18.103');">18.103</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.103" title="18.103" class="bubblelink code" onclick="return showCourse(this, '18.103');">18.103</a>. </p> <p class="courseblockinstructors seemore"><span><i>J. Shi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.104 Seminar in Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a> </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Students present and discuss material from books or journals. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>Fall: T. Ozuch-Meersseman. Spring: G. Staffilani</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.112 Functions of a Complex Variable</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.1121" title="18.1121" class="bubblelink code" onclick="return showCourse(this, '18.1121');">18.1121</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions. </p> <p class="courseblockinstructors seemore"><span><i>A. Lawrie</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.1121 Functions of a Complex Variable</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.112" title="18.112" class="bubblelink code" onclick="return showCourse(this, '18.112');">18.112</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.112" title="18.112" class="bubblelink code" onclick="return showCourse(this, '18.112');">18.112</a>. </p> <p class="courseblockinstructors seemore"><span><i>A. Lawrie</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.116 Riemann Surfaces</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.112" title="18.112" class="bubblelink code" onclick="return showCourse(this, '18.112');">18.112</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Riemann surfaces, uniformization, Riemann-Roch Theorem. Theory of elliptic functions and modular forms. Some applications, such as to number theory. </p> <p class="courseblockinstructors seemore"><span><i>P. I. Etingof</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.117 Topics in Several Complex Variables</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.112" title="18.112" class="bubblelink code" onclick="return showCourse(this, '18.112');">18.112</a> and <a href="/search/?P=18.965" title="18.965" class="bubblelink code" onclick="return showCourse(this, '18.965');">18.965</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Harmonic theory on complex manifolds, Hodge decomposition theorem, Hard Lefschetz theorem. Vanishing theorems. Theory of Stein manifolds. As time permits students also study holomorphic vector bundles on Kahler manifolds. </p> <p class="courseblockinstructors seemore"><span><i>B. Poonen</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.118 Topics in Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>S. Dyatlov</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.125 Measure Theory and Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Provides a rigorous introduction to Lebesgue's theory of measure and integration. Covers material that is essential in analysis, probability theory, and differential geometry. </p> <p class="courseblockinstructors seemore"><span><i>M. Jezequel</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.137 Topics in Geometric Partial Differential Equations</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Fall)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>T. Colding</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.152 Introduction to Partial Differential Equations</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.1521" title="18.1521" class="bubblelink code" onclick="return showCourse(this, '18.1521');">18.1521</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems. </p> <p class="courseblockinstructors seemore"><span><i>T. Collins</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.1521 Introduction to Partial Differential Equations</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.152" title="18.152" class="bubblelink code" onclick="return showCourse(this, '18.152');">18.152</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.152" title="18.152" class="bubblelink code" onclick="return showCourse(this, '18.152');">18.152</a>. </p> <p class="courseblockinstructors seemore"><span><i>T. Collins</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.155 Differential Analysis I</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.102" title="18.102" class="bubblelink code" onclick="return showCourse(this, '18.102');">18.102</a> or <a href="/search/?P=18.103" title="18.103" class="bubblelink code" onclick="return showCourse(this, '18.103');">18.103</a> </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">First part of a two-subject sequence. Review of Lebesgue integration. Lp spaces. Distributions. Fourier transform. Sobolev spaces. Spectral theorem, discrete and continuous spectrum. Homogeneous distributions. Fundamental solutions for elliptic, hyperbolic and parabolic differential operators. Recommended prerequisite: <a href="/search/?P=18.112" title="18.112" class="bubblelink code" onclick="return showCourse(this, '18.112');">18.112</a>. </p> <p class="courseblockinstructors seemore"><span><i>S. Dyatlov</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.156 Differential Analysis II</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.155" title="18.155" class="bubblelink code" onclick="return showCourse(this, '18.155');">18.155</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Second part of a two-subject sequence. Covers variable coefficient elliptic, parabolic and hyperbolic partial differential equations. </p> <p class="courseblockinstructors seemore"><span><i>D. Jerison</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.157 Introduction to Microlocal Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.155" title="18.155" class="bubblelink code" onclick="return showCourse(this, '18.155');">18.155</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">The semi-classical theory of partial differential equations. Discussion of Pseudodifferential operators, Fourier integral operators, asymptotic solutions of partial differential equations, and the spectral theory of Schroedinger operators from the semi-classical perspective. Heavy emphasis placed on the symplectic geometric underpinnings of this subject. </p> <p class="courseblockinstructors seemore"><span><i>R. B. Melrose</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.158 Topics in Differential Equations</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.157" title="18.157" class="bubblelink code" onclick="return showCourse(this, '18.157');">18.157</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>L. Guth</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.199 Graduate Analysis Seminar</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Studies original papers in differential analysis and differential equations. Intended for first- and second-year graduate students. Permission must be secured in advance. </p> <p class="courseblockinstructors seemore"><span><i>V. W. Guillemin</i></span></p> </div><h2 class="sectionhead">Discrete Applied Mathematics</h2><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.200 Principles of Discrete Applied Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None. <i>Coreq: <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a></i> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">4-0-11 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.200A" title="18.200A" class="bubblelink code" onclick="return showCourse(this, '18.200A');">18.200A</a></span> </p> <p class="courseblockdesc">Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. Instruction and practice in written communication provided. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>P. W. Shor, A. Moitra</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.200A Principles of Discrete Applied Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None. <i>Coreq: <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a></i> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=18.200" title="18.200" class="bubblelink code" onclick="return showCourse(this, '18.200');">18.200</a></span> </p> <p class="courseblockdesc">Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.204 Undergraduate Seminar in Discrete Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: ((<a href="/search/?P=6.1200" title="6.1200[J]" class="bubblelink code" onclick="return showCourse(this, '6.1200');">6.1200[J]</a> or <a href="/search/?P=18.200" title="18.200" class="bubblelink code" onclick="return showCourse(this, '18.200');">18.200</a>) and (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>)) or permission of instructor </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Seminar in combinatorics, graph theory, and discrete mathematics in general. Participants read and present papers from recent mathematics literature. Instruction and practice in written and oral communication provided. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>J. He, D. Mikulincer, M. Sherman-Bennett, A. Weigandt</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.211 Combinatorial Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Combinatorial problems and methods for their solution. Enumeration, generating functions, recurrence relations, construction of bijections. Introduction to graph theory. Prior experience with abstraction and proofs is helpful. </p> <p class="courseblockinstructors seemore"><span><i>A. Weigandt</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.212 Algebraic Combinatorics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> or <a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Applications of algebra to combinatorics. Topics include walks in graphs, the Radon transform, groups acting on posets, Young tableaux, electrical networks. </p> <p class="courseblockinstructors seemore"><span><i>A. Postnikov</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.217 Combinatorial Theory</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Content varies from year to year. </p> <p class="courseblockinstructors seemore"><span><i>A. Postnikov</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.218 Topics in Combinatorics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>L. Sauermann</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.219 Seminar in Combinatorics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Fall)<br/> Not offered regularly; consult department</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Content varies from year to year. Readings from current research papers in combinatorics. Topics to be chosen and presented by the class. </p> <p class="courseblockinstructors seemore"><span><i>Information: Y. Zhao</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.225 Graph Theory and Additive Combinatorics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: ((<a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> or <a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>)) or permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to extremal graph theory and additive combinatorics. Highlights common themes, such as the dichotomy between structure versus pseudorandomness. Topics include Turan-type problems, Szemeredi's regularity lemma and applications, pseudorandom graphs, spectral graph theory, graph limits, arithmetic progressions (Roth, Szemeredi, Green-Tao), discrete Fourier analysis, Freiman's theorem on sumsets and structure. Discusses current research topics and open problems. </p> <p class="courseblockinstructors seemore"><span><i>Y. Zhao</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.226 Probabilistic Methods in Combinatorics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: (<a href="/search/?P=18.211" title="18.211" class="bubblelink code" onclick="return showCourse(this, '18.211');">18.211</a>, <a href="/search/?P=18.600" title="18.600" class="bubblelink code" onclick="return showCourse(this, '18.600');">18.600</a>, and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>)) or permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Fall)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to the probabilistic method, a fundamental and powerful technique in combinatorics and theoretical computer science. Focuses on methodology as well as combinatorial applications. Suitable for students with strong interest and background in mathematical problem solving. Topics include linearity of expectations, alteration, second moment, Lovasz local lemma, correlation inequalities, Janson inequalities, concentration inequalities, entropy method. </p> <p class="courseblockinstructors seemore"><span><i>Y. Zhao</i></span></p> </div><h2 class="sectionhead">Continuous Applied Mathematics</h2><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.300 Principles of Continuum Applied Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Covers fundamental concepts in continuous applied mathematics. Applications from traffic flow, fluids, elasticity, granular flows, etc. Also covers continuum limit; conservation laws, quasi-equilibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion and group velocity. Uses MATLAB computing environment. </p> <p class="courseblockinstructors seemore"><span><i>B. Geshkovski</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.303 Linear Partial Differential Equations: Analysis and Numerics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a> or <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science and engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Studies operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit timestepping. Some programming required for homework and final project. </p> <p class="courseblockinstructors seemore"><span><i>V. Heinonen</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.305 Advanced Analytic Methods in Science and Engineering</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.04" title="18.04" class="bubblelink code" onclick="return showCourse(this, '18.04');">18.04</a>, <a href="/search/?P=18.075" title="18.075" class="bubblelink code" onclick="return showCourse(this, '18.075');">18.075</a>, or <a href="/search/?P=18.112" title="18.112" class="bubblelink code" onclick="return showCourse(this, '18.112');">18.112</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Covers expansion around singular points: the WKB method on ordinary and partial differential equations; the method of stationary phase and the saddle point method; the two-scale method and the method of renormalized perturbation; singular perturbation and boundary-layer techniques; WKB method on partial differential equations. </p> <p class="courseblockinstructors seemore"><span><i>H. Cheng</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.306 Advanced Partial Differential Equations with Applications</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) and (<a href="/search/?P=18.04" title="18.04" class="bubblelink code" onclick="return showCourse(this, '18.04');">18.04</a>, <a href="/search/?P=18.075" title="18.075" class="bubblelink code" onclick="return showCourse(this, '18.075');">18.075</a>, or <a href="/search/?P=18.112" title="18.112" class="bubblelink code" onclick="return showCourse(this, '18.112');">18.112</a>) </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Concepts and techniques for partial differential equations, especially nonlinear. Diffusion, dispersion and other phenomena. Initial and boundary value problems. Normal mode analysis, Green's functions, and transforms. Conservation laws, kinematic waves, hyperbolic equations, characteristics shocks, simple waves. Geometrical optics, caustics. Free-boundary problems. Dimensional analysis. Singular perturbation, boundary layers, homogenization. Variational methods. Solitons. Applications from fluid dynamics, materials science, optics, traffic flow, etc. </p> <p class="courseblockinstructors seemore"><span><i>R. R. Rosales</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.327 Topics in Applied Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>L. Demanet</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.330 Introduction to Numerical Analysis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Basic techniques for the efficient numerical solution of problems in science and engineering. Root finding, interpolation, approximation of functions, integration, differential equations, direct and iterative methods in linear algebra. Knowledge of programming in a language such as MATLAB, Python, or Julia is helpful. </p> <p class="courseblockinstructors seemore"><span><i>L. Demanet</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.335[J] Introduction to Numerical Methods</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.7310J" title="6.7310[J]" class="bubblelink code" onclick="return showCourse(this, '6.7310J');">6.7310[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Advanced introduction to numerical analysis: accuracy and efficiency of numerical algorithms. In-depth coverage of sparse-matrix/iterative and dense-matrix algorithms in numerical linear algebra (for linear systems and eigenproblems). Floating-point arithmetic, backwards error analysis, conditioning, and stability. Other computational topics (e.g., numerical integration or nonlinear optimization) may also be surveyed. Final project involves some programming. </p> <p class="courseblockinstructors seemore"><span><i>A. J. Horning</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.336[J] Fast Methods for Partial Differential and Integral Equations</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.7340J" title="6.7340[J]" class="bubblelink code" onclick="return showCourse(this, '6.7340J');">6.7340[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.7300" title="6.7300[J]" class="bubblelink code" onclick="return showCourse(this, '6.7300');">6.7300[J]</a>, <a href="/search/?P=16.920" title="16.920[J]" class="bubblelink code" onclick="return showCourse(this, '16.920');">16.920[J]</a>, <a href="/search/?P=18.085" title="18.085" class="bubblelink code" onclick="return showCourse(this, '18.085');">18.085</a>, <a href="/search/?P=18.335" title="18.335[J]" class="bubblelink code" onclick="return showCourse(this, '18.335');">18.335[J]</a>, or permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall, Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Unified introduction to the theory and practice of modern, near linear-time, numerical methods for large-scale partial-differential and integral equations. Topics include preconditioned iterative methods; generalized Fast Fourier Transform and other butterfly-based methods; multiresolution approaches, such as multigrid algorithms and hierarchical low-rank matrix decompositions; and low and high frequency Fast Multipole Methods. Example applications include aircraft design, cardiovascular system modeling, electronic structure computation, and tomographic imaging. </p> <p class="courseblockinstructors seemore"><span><i>K. Burns</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.337[J] Parallel Computing and Scientific Machine Learning</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.7320J" title="6.7320[J]" class="bubblelink code" onclick="return showCourse(this, '6.7320J');">6.7320[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to scientific machine learning with an emphasis on developing scalable differentiable programs. Covers scientific computing topics (numerical differential equations, dense and sparse linear algebra, Fourier transformations, parallelization of large-scale scientific simulation) simultaneously with modern data science (machine learning, deep neural networks, automatic differentiation), focusing on the emerging techniques at the connection between these areas, such as neural differential equations and physics-informed deep learning. Provides direct experience with the modern realities of optimizing code performance for supercomputers, GPUs, and multicores in a high-level language. </p> <p class="courseblockinstructors seemore"><span><i>A. Edelman</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.338 Eigenvalues of Random Matrices</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> or permission of instructor </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Covers the modern main results of random matrix theory as it is currently applied in engineering and science. Topics include matrix calculus for finite and infinite matrices (e.g., Wigner's semi-circle and Marcenko-Pastur laws), free probability, random graphs, combinatorial methods, matrix statistics, stochastic operators, passage to the continuum limit, moment methods, and compressed sensing. Knowledge of Julia helpful, but not required. </p> <p class="courseblockinstructors seemore"><span><i>A. Edelman</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.352[J] Nonlinear Dynamics: The Natural Environment</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=12.009J" title="12.009[J]" class="bubblelink code" onclick="return showCourse(this, '12.009J');">12.009[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> and <a href="/search/?P=8.01|8.01L|8.011|8.012" onclick="return showCourse(this, '8.01|8.01L|8.011|8.012');">Physics I (GIR)</a>; <i>Coreq: <a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a></i> </span><br/><span class="courseblockterms">U (Fall)<br/> Not offered regularly; consult department</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=12.009J" title="12.009[J]" class="bubblelink code" onclick="return showCourse(this, '12.009J');">12.009[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>D. H. Rothman</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.353[J] Nonlinear Dynamics: Chaos</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=2.050J" title="2.050[J]" class="bubblelink code" onclick="return showCourse(this, '2.050J');">2.050[J]</a>, <a href="/search/?P=12.006J" title="12.006[J]" class="bubblelink code" onclick="return showCourse(this, '12.006J');">12.006[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=8.02|8.021|8.022" onclick="return showCourse(this, '8.02|8.021|8.022');">Physics II (GIR)</a> and (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=12.006J" title="12.006[J]" class="bubblelink code" onclick="return showCourse(this, '12.006J');">12.006[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>D. Rothman</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.354[J] Nonlinear Dynamics: Continuum Systems</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=1.062J" title="1.062[J]" class="bubblelink code" onclick="return showCourse(this, '1.062J');">1.062[J]</a>, <a href="/search/?P=12.207J" title="12.207[J]" class="bubblelink code" onclick="return showCourse(this, '12.207J');">12.207[J]</a><br/> Subject meets with <a href="/search/?P=18.3541" title="18.3541" class="bubblelink code" onclick="return showCourse(this, '18.3541');">18.3541</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=8.02|8.021|8.022" onclick="return showCourse(this, '8.02|8.021|8.022');">Physics II (GIR)</a> and (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">General mathematical principles of continuum systems. From microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) differential equations. Exact solutions, dimensional analysis, calculus of variations and singular perturbation methods. Stability, waves and pattern formation in continuum systems. Subject matter illustrated using natural fluid and solid systems found, for example, in geophysics and biology. </p> <p class="courseblockinstructors seemore"><span><i>B. Primkulov</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.3541 Nonlinear Dynamics: Continuum Systems</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=1.062J" title="1.062[J]" class="bubblelink code" onclick="return showCourse(this, '1.062J');">1.062[J]</a>, <a href="/search/?P=12.207J" title="12.207[J]" class="bubblelink code" onclick="return showCourse(this, '12.207J');">12.207[J]</a>, <a href="/search/?P=18.354J" title="18.354[J]" class="bubblelink code" onclick="return showCourse(this, '18.354J');">18.354[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=8.02|8.021|8.022" onclick="return showCourse(this, '8.02|8.021|8.022');">Physics II (GIR)</a> and (<a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a>) </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">General mathematical principles of continuum systems. From microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) differential equations. Exact solutions, dimensional analysis, calculus of variations and singular perturbation methods. Stability, waves and pattern formation in continuum systems. Subject matter illustrated using natural fluid and solid systems found, for example, in geophysics and biology. Students in Courses 1, 12, and 18 must register for undergraduate version, <a href="/search/?P=18.354" title="18.354[J]" class="bubblelink code" onclick="return showCourse(this, '18.354');">18.354[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>B. Primkulov</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.355 Fluid Mechanics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=2.25" title="2.25" class="bubblelink code" onclick="return showCourse(this, '2.25');">2.25</a>, <a href="/search/?P=12.800" title="12.800" class="bubblelink code" onclick="return showCourse(this, '12.800');">12.800</a>, or <a href="/search/?P=18.354" title="18.354[J]" class="bubblelink code" onclick="return showCourse(this, '18.354');">18.354[J]</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Topics include the development of Navier-Stokes equations, inviscid flows, boundary layers, lubrication theory, Stokes flows, and surface tension. Fundamental concepts illustrated through problems drawn from a variety of areas, including geophysics, biology, and the dynamics of sport. Particular emphasis on the interplay between dimensional analysis, scaling arguments, and theory. Includes classroom and laboratory demonstrations. </p> <p class="courseblockinstructors seemore"><span><i>J. W. Bush</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.357 Interfacial Phenomena</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=2.25" title="2.25" class="bubblelink code" onclick="return showCourse(this, '2.25');">2.25</a>, <a href="/search/?P=12.800" title="12.800" class="bubblelink code" onclick="return showCourse(this, '12.800');">12.800</a>, <a href="/search/?P=18.354" title="18.354[J]" class="bubblelink code" onclick="return showCourse(this, '18.354');">18.354[J]</a>, <a href="/search/?P=18.355" title="18.355" class="bubblelink code" onclick="return showCourse(this, '18.355');">18.355</a>, or permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Fluid systems dominated by the influence of interfacial tension. Elucidates the roles of curvature pressure and Marangoni stress in a variety of hydrodynamic settings. Particular attention to drops and bubbles, soap films and minimal surfaces, wetting phenomena, water-repellency, surfactants, Marangoni flows, capillary origami and contact line dynamics. Theoretical developments are accompanied by classroom demonstrations. Highlights the role of surface tension in biology. </p> <p class="courseblockinstructors seemore"><span><i>J. W. Bush</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.358[J] Nonlinear Dynamics and Turbulence</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=1.686J" title="1.686[J]" class="bubblelink code" onclick="return showCourse(this, '1.686J');">1.686[J]</a>, <a href="/search/?P=2.033J" title="2.033[J]" class="bubblelink code" onclick="return showCourse(this, '2.033J');">2.033[J]</a><br/> Subject meets with <a href="/search/?P=1.068" title="1.068" class="bubblelink code" onclick="return showCourse(this, '1.068');">1.068</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=1.060A" title="1.060A" class="bubblelink code" onclick="return showCourse(this, '1.060A');">1.060A</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-2-7 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=1.686J" title="1.686[J]" class="bubblelink code" onclick="return showCourse(this, '1.686J');">1.686[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>L. Bourouiba</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.367 Waves and Imaging</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Fall)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">The mathematics of inverse problems involving waves, with examples taken from reflection seismology, synthetic aperture radar, and computerized tomography. Suitable for graduate students from all departments who have affinities with applied mathematics. Topics include acoustic, elastic, electromagnetic wave equations; geometrical optics; scattering series and inversion; migration and backprojection; adjoint-state methods; Radon and curvilinear Radon transforms; microlocal analysis of imaging; optimization, regularization, and sparse regression. </p> <p class="courseblockinstructors seemore"><span><i>L. Demanet</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.369[J] Mathematical Methods in Nanophotonics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=8.315J" title="8.315[J]" class="bubblelink code" onclick="return showCourse(this, '8.315J');">8.315[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=8.07" title="8.07" class="bubblelink code" onclick="return showCourse(this, '8.07');">8.07</a>, <a href="/search/?P=18.303" title="18.303" class="bubblelink code" onclick="return showCourse(this, '18.303');">18.303</a>, or permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">High-level approaches to understanding complex optical media, structured on the scale of the wavelength, that are not generally analytically soluable. The basis for understanding optical phenomena such as photonic crystals and band gaps, anomalous diffraction, mechanisms for optical confinement, optical fibers (new and old), nonlinearities, and integrated optical devices. Methods covered include linear algebra and eigensystems for Maxwell's equations, symmetry groups and representation theory, Bloch's theorem, numerical eigensolver methods, time and frequency-domain computation, perturbation theory, and coupled-mode theories. </p> <p class="courseblockinstructors seemore"><span><i>S. G. Johnson</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.376[J] Wave Propagation</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=1.138J" title="1.138[J]" class="bubblelink code" onclick="return showCourse(this, '1.138J');">1.138[J]</a>, <a href="/search/?P=2.062J" title="2.062[J]" class="bubblelink code" onclick="return showCourse(this, '2.062J');">2.062[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=2.003" title="2.003[J]" class="bubblelink code" onclick="return showCourse(this, '2.003');">2.003[J]</a> and <a href="/search/?P=18.075" title="18.075" class="bubblelink code" onclick="return showCourse(this, '18.075');">18.075</a> </span><br/><span class="courseblockterms">G (Spring)<br/> Not offered regularly; consult department</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=2.062J" title="2.062[J]" class="bubblelink code" onclick="return showCourse(this, '2.062J');">2.062[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>T. R. Akylas, R. R. Rosales</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.377[J] Nonlinear Dynamics and Waves</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=1.685J" title="1.685[J]" class="bubblelink code" onclick="return showCourse(this, '1.685J');">1.685[J]</a>, <a href="/search/?P=2.034J" title="2.034[J]" class="bubblelink code" onclick="return showCourse(this, '2.034J');">2.034[J]</a></span><br/><span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">A unified treatment of nonlinear oscillations and wave phenomena with applications to mechanical, optical, geophysical, fluid, electrical and flow-structure interaction problems. Nonlinear free and forced vibrations; nonlinear resonances; self-excited oscillations; lock-in phenomena. Nonlinear dispersive and nondispersive waves; resonant wave interactions; propagation of wave pulses and nonlinear Schrodinger equation. Nonlinear long waves and breaking; theory of characteristics; the Korteweg-de Vries equation; solitons and solitary wave interactions. Stability of shear flows. Some topics and applications may vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>R. R. Rosales</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.384 Undergraduate Seminar in Physical Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=12.006" title="12.006[J]" class="bubblelink code" onclick="return showCourse(this, '12.006');">12.006[J]</a>, <a href="/search/?P=18.300" title="18.300" class="bubblelink code" onclick="return showCourse(this, '18.300');">18.300</a>, <a href="/search/?P=18.354" title="18.354[J]" class="bubblelink code" onclick="return showCourse(this, '18.354');">18.354[J]</a>, or permission of instructor </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Covers the mathematical modeling of physical systems, with emphasis on the reading and presentation of papers. Addresses a broad range of topics, with particular focus on macroscopic physics and continuum systems: fluid dynamics, solid mechanics, and biophysics. Instruction and practice in written and oral communication provided. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>O. Kodio</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.385[J] Nonlinear Dynamics and Chaos</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=2.036J" title="2.036[J]" class="bubblelink code" onclick="return showCourse(this, '2.036J');">2.036[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.032" title="18.032" class="bubblelink code" onclick="return showCourse(this, '18.032');">18.032</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to the theory of nonlinear dynamical systems with applications from science and engineering. Local and global existence of solutions, dependence on initial data and parameters. Elementary bifurcations, normal forms. Phase plane, limit cycles, relaxation oscillations, Poincare-Bendixson theory. Floquet theory. Poincare maps. Averaging. Near-equilibrium dynamics. Synchronization. Introduction to chaos. Universality. Strange attractors. Lorenz and Rossler systems. Hamiltonian dynamics and KAM theory. Uses MATLAB computing environment. </p> <p class="courseblockinstructors seemore"><span><i>R. R. Rosales</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.397 Mathematical Methods in Physics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.745" title="18.745" class="bubblelink code" onclick="return showCourse(this, '18.745');">18.745</a> or some familiarity with Lie theory </span><br/><span class="courseblockterms">G (Fall)<br/> Not offered regularly; consult department</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Content varies from year to year. Recent developments in quantum field theory require mathematical techniques not usually covered in standard graduate subjects. </p> <p class="courseblockinstructors seemore"><span><i>V. G. Kac</i></span></p> </div><h2 class="sectionhead">Theoretical Computer Science</h2><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.400[J] Computability and Complexity Theory</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.1400J" title="6.1400[J]" class="bubblelink code" onclick="return showCourse(this, '6.1400J');">6.1400[J]</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=6.1200" title="6.1200[J]" class="bubblelink code" onclick="return showCourse(this, '6.1200');">6.1200[J]</a> and <a href="/search/?P=6.1210" title="6.1210" class="bubblelink code" onclick="return showCourse(this, '6.1210');">6.1210</a>) or permission of instructor </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">4-0-8 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=6.1400J" title="6.1400[J]" class="bubblelink code" onclick="return showCourse(this, '6.1400J');">6.1400[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>R. Williams, R. Rubinfeld</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.404 Theory of Computation</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=6.5400J" title="6.5400[J]" class="bubblelink code" onclick="return showCourse(this, '6.5400J');">6.5400[J]</a>, <a href="/search/?P=18.4041J" title="18.4041[J]" class="bubblelink code" onclick="return showCourse(this, '18.4041J');">18.4041[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.1200" title="6.1200[J]" class="bubblelink code" onclick="return showCourse(this, '6.1200');">6.1200[J]</a> or <a href="/search/?P=18.200" title="18.200" class="bubblelink code" onclick="return showCourse(this, '18.200');">18.200</a> </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">4-0-8 units</span> </p> <p class="courseblockdesc">A more extensive and theoretical treatment of the material in <a href="/search/?P=6.1400J" title="6.1400[J]" class="bubblelink code" onclick="return showCourse(this, '6.1400J');">6.1400[J]</a>/<a href="/search/?P=18.400J" title="18.400[J]" class="bubblelink code" onclick="return showCourse(this, '18.400J');">18.400[J]</a>, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. </p> <p class="courseblockinstructors seemore"><span><i>M. Sipser</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.4041[J] Theory of Computation</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.5400J" title="6.5400[J]" class="bubblelink code" onclick="return showCourse(this, '6.5400J');">6.5400[J]</a><br/> Subject meets with <a href="/search/?P=18.404" title="18.404" class="bubblelink code" onclick="return showCourse(this, '18.404');">18.404</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.1200" title="6.1200[J]" class="bubblelink code" onclick="return showCourse(this, '6.1200');">6.1200[J]</a> or <a href="/search/?P=18.200" title="18.200" class="bubblelink code" onclick="return showCourse(this, '18.200');">18.200</a> </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">4-0-8 units</span> </p> <p class="courseblockdesc">A more extensive and theoretical treatment of the material in <a href="/search/?P=6.1400J" title="6.1400[J]" class="bubblelink code" onclick="return showCourse(this, '6.1400J');">6.1400[J]</a>/<a href="/search/?P=18.400J" title="18.400[J]" class="bubblelink code" onclick="return showCourse(this, '18.400J');">18.400[J]</a>, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.404" title="18.404" class="bubblelink code" onclick="return showCourse(this, '18.404');">18.404</a>. </p> <p class="courseblockinstructors seemore"><span><i>M. Sipser</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.405[J] Advanced Complexity Theory</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.5410J" title="6.5410[J]" class="bubblelink code" onclick="return showCourse(this, '6.5410J');">6.5410[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.404" title="18.404" class="bubblelink code" onclick="return showCourse(this, '18.404');">18.404</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Current research topics in computational complexity theory. Nondeterministic, alternating, probabilistic, and parallel computation models. Boolean circuits. Complexity classes and complete sets. The polynomial-time hierarchy. Interactive proof systems. Relativization. Definitions of randomness. Pseudo-randomness and derandomizations. Interactive proof systems and probabilistically checkable proofs. </p> <p class="courseblockinstructors seemore"><span><i>R. Williams</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.408 Topics in Theoretical Computer Science</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Fall)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Study of areas of current interest in theoretical computer science. Topics vary from term to term. </p> <p class="courseblockinstructors seemore"><span><i>Fall: D. Minzer. Spring: A. Moitra</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.410[J] Design and Analysis of Algorithms</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.1220J" title="6.1220[J]" class="bubblelink code" onclick="return showCourse(this, '6.1220J');">6.1220[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.1200" title="6.1200[J]" class="bubblelink code" onclick="return showCourse(this, '6.1200');">6.1200[J]</a> and <a href="/search/?P=6.1210" title="6.1210" class="bubblelink code" onclick="return showCourse(this, '6.1210');">6.1210</a> </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">4-0-8 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=6.1220J" title="6.1220[J]" class="bubblelink code" onclick="return showCourse(this, '6.1220J');">6.1220[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>E. Demaine, M. Goemans, S. Raghuraman</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.413 Introduction to Computational Molecular Biology</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.417" title="18.417" class="bubblelink code" onclick="return showCourse(this, '18.417');">18.417</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.1210" title="6.1210" class="bubblelink code" onclick="return showCourse(this, '6.1210');">6.1210</a> or permission of instructor </span><br/><span class="courseblockterms">U (Spring)<br/> Not offered regularly; consult department</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to computational molecular biology with a focus on the basic computational algorithms used to solve problems in practice. Covers classical techniques in the field for solving problems such as genome sequencing, assembly, and search; detecting genome rearrangements; constructing evolutionary trees; analyzing mass spectrometry data; connecting gene expression to cellular function; and machine learning for drug discovery. Prior knowledge of biology is not required. Particular emphasis on problem solving, collaborative learning, theoretical analysis, and practical implementation of algorithms. Students taking graduate version complete additional and more complex assignments. </p> <p class="courseblockinstructors seemore"><span><i>B. Berger</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.415[J] Advanced Algorithms</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.5210J" title="6.5210[J]" class="bubblelink code" onclick="return showCourse(this, '6.5210J');">6.5210[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.1220" title="6.1220[J]" class="bubblelink code" onclick="return showCourse(this, '6.1220');">6.1220[J]</a> and (<a href="/search/?P=6.1200" title="6.1200[J]" class="bubblelink code" onclick="return showCourse(this, '6.1200');">6.1200[J]</a>, <a href="/search/?P=6.3700" title="6.3700" class="bubblelink code" onclick="return showCourse(this, '6.3700');">6.3700</a>, or <a href="/search/?P=18.600" title="18.600" class="bubblelink code" onclick="return showCourse(this, '18.600');">18.600</a>) </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">5-0-7 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=6.5210J" title="6.5210[J]" class="bubblelink code" onclick="return showCourse(this, '6.5210J');">6.5210[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>A. Moitra, D. R. Karger</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.416[J] Randomized Algorithms</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.5220J" title="6.5220[J]" class="bubblelink code" onclick="return showCourse(this, '6.5220J');">6.5220[J]</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=6.1200" title="6.1200[J]" class="bubblelink code" onclick="return showCourse(this, '6.1200');">6.1200[J]</a> or <a href="/search/?P=6.3700" title="6.3700" class="bubblelink code" onclick="return showCourse(this, '6.3700');">6.3700</a>) and (<a href="/search/?P=6.1220" title="6.1220[J]" class="bubblelink code" onclick="return showCourse(this, '6.1220');">6.1220[J]</a> or <a href="/search/?P=6.5210" title="6.5210[J]" class="bubblelink code" onclick="return showCourse(this, '6.5210');">6.5210[J]</a>) </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">5-0-7 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=6.5220J" title="6.5220[J]" class="bubblelink code" onclick="return showCourse(this, '6.5220J');">6.5220[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>D. R. Karger</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.417 Introduction to Computational Molecular Biology</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.413" title="18.413" class="bubblelink code" onclick="return showCourse(this, '18.413');">18.413</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.1210" title="6.1210" class="bubblelink code" onclick="return showCourse(this, '6.1210');">6.1210</a> or permission of instructor </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to computational molecular biology with a focus on the basic computational algorithms used to solve problems in practice. Covers classical techniques in the field for solving problems such as genome sequencing, assembly, and search; detecting genome rearrangements; constructing evolutionary trees; analyzing mass spectrometry data; connecting gene expression to cellular function; and machine learning for drug discovery. Prior knowledge of biology is not required. Particular emphasis on problem solving, collaborative learning, theoretical analysis, and practical implementation of algorithms. Students taking graduate version complete additional and more complex assignments. </p> <p class="courseblockinstructors seemore"><span><i>B. Berger</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.418[J] Topics in Computational Molecular Biology</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=HST.504J" title="HST.504[J]" class="bubblelink code" onclick="return showCourse(this, 'HST.504J');">HST.504[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.8701" title="6.8701" class="bubblelink code" onclick="return showCourse(this, '6.8701');">6.8701</a>, <a href="/search/?P=18.417" title="18.417" class="bubblelink code" onclick="return showCourse(this, '18.417');">18.417</a>, or permission of instructor </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Covers current research topics in computational molecular biology. Recent research papers presented from leading conferences such as the International Conference on Computational Molecular Biology (RECOMB) and the Conference on Intelligent Systems for Molecular Biology (ISMB). Topics include original research (both theoretical and experimental) in comparative genomics, sequence and structure analysis, molecular evolution, proteomics, gene expression, transcriptional regulation, biological networks, drug discovery, and privacy. Recent research by course participants also covered. Participants will be expected to present individual projects to the class. </p> <p class="courseblockinstructors seemore"><span><i>B. Berger</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.424 Seminar in Information Theory</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: (<a href="/search/?P=6.3700" title="6.3700" class="bubblelink code" onclick="return showCourse(this, '6.3700');">6.3700</a>, <a href="/search/?P=18.05" title="18.05" class="bubblelink code" onclick="return showCourse(this, '18.05');">18.05</a>, or <a href="/search/?P=18.600" title="18.600" class="bubblelink code" onclick="return showCourse(this, '18.600');">18.600</a>) and (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Considers various topics in information theory, including data compression, Shannon's Theorems, and error-correcting codes. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>J. Kelner</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.425[J] Foundations of Cryptography</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.5620J" title="6.5620[J]" class="bubblelink code" onclick="return showCourse(this, '6.5620J');">6.5620[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.1220" title="6.1220[J]" class="bubblelink code" onclick="return showCourse(this, '6.1220');">6.1220[J]</a>, <a href="/search/?P=6.1400" title="6.1400[J]" class="bubblelink code" onclick="return showCourse(this, '6.1400');">6.1400[J]</a>, or <a href="/search/?P=18.4041" title="18.4041[J]" class="bubblelink code" onclick="return showCourse(this, '18.4041');">18.4041[J]</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=6.5620J" title="6.5620[J]" class="bubblelink code" onclick="return showCourse(this, '6.5620J');">6.5620[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>S. Goldwasser, S. Micali, V. Vaikuntanathan</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.434 Seminar in Theoretical Computer Science</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=6.1220" title="6.1220[J]" class="bubblelink code" onclick="return showCourse(this, '6.1220');">6.1220[J]</a> </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>Fall: E. Mossel. Spring: D. Minzer</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.435[J] Quantum Computation</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=2.111J" title="2.111[J]" class="bubblelink code" onclick="return showCourse(this, '2.111J');">2.111[J]</a>, <a href="/search/?P=6.6410J" title="6.6410[J]" class="bubblelink code" onclick="return showCourse(this, '6.6410J');">6.6410[J]</a>, <a href="/search/?P=8.370J" title="8.370[J]" class="bubblelink code" onclick="return showCourse(this, '8.370J');">8.370[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=8.05" title="8.05" class="bubblelink code" onclick="return showCourse(this, '8.05');">8.05</a>, <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>, or <a href="/search/?P=18.C06" title="18.C06[J]" class="bubblelink code" onclick="return showCourse(this, '18.C06');">18.C06[J]</a> </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Provides an introduction to the theory and practice of quantum computation. Topics covered: physics of information processing; quantum algorithms including the factoring algorithm and Grover's search algorithm; quantum error correction; quantum communication and cryptography. Knowledge of quantum mechanics helpful but not required. </p> <p class="courseblockinstructors seemore"><span><i>I. Chuang, A. Harrow, P. Shor</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.436[J] Quantum Information Science</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.6420J" title="6.6420[J]" class="bubblelink code" onclick="return showCourse(this, '6.6420J');">6.6420[J]</a>, <a href="/search/?P=8.371J" title="8.371[J]" class="bubblelink code" onclick="return showCourse(this, '8.371J');">8.371[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.435" title="18.435[J]" class="bubblelink code" onclick="return showCourse(this, '18.435');">18.435[J]</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=8.371J" title="8.371[J]" class="bubblelink code" onclick="return showCourse(this, '8.371J');">8.371[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>I. Chuang, A. Harrow</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.437[J] Distributed Algorithms</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.5250J" title="6.5250[J]" class="bubblelink code" onclick="return showCourse(this, '6.5250J');">6.5250[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.1220" title="6.1220[J]" class="bubblelink code" onclick="return showCourse(this, '6.1220');">6.1220[J]</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=6.5250J" title="6.5250[J]" class="bubblelink code" onclick="return showCourse(this, '6.5250J');">6.5250[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>M. Ghaffari, N. A. Lynch</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.453 Combinatorial Optimization</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.4531" title="18.4531" class="bubblelink code" onclick="return showCourse(this, '18.4531');">18.4531</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Thorough treatment of linear programming and combinatorial optimization. Topics include matching theory, network flow, matroid optimization, and how to deal with NP-hard optimization problems. Prior exposure to discrete mathematics (such as <a href="/search/?P=18.200" title="18.200" class="bubblelink code" onclick="return showCourse(this, '18.200');">18.200</a>) helpful. </p> <p class="courseblockinstructors seemore"><span><i>Information: M. X. Goemans</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.4531 Combinatorial Optimization</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.453" title="18.453" class="bubblelink code" onclick="return showCourse(this, '18.453');">18.453</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Thorough treatment of linear programming and combinatorial optimization. Topics include matching theory, network flow, matroid optimization, and how to deal with NP-hard optimization problems. Prior exposure to discrete mathematics (such as <a href="/search/?P=18.200" title="18.200" class="bubblelink code" onclick="return showCourse(this, '18.200');">18.200</a>) helpful. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.453" title="18.453" class="bubblelink code" onclick="return showCourse(this, '18.453');">18.453</a>. </p> <p class="courseblockinstructors seemore"><span><i>Information: M. X. Goemans</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.455 Advanced Combinatorial Optimization</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.453" title="18.453" class="bubblelink code" onclick="return showCourse(this, '18.453');">18.453</a> or permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Advanced treatment of combinatorial optimization with an emphasis on combinatorial aspects. Non-bipartite matchings, submodular functions, matroid intersection/union, matroid matching, submodular flows, multicommodity flows, packing and connectivity problems, and other recent developments. </p> <p class="courseblockinstructors seemore"><span><i>M. X. Goemans</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.456[J] Algebraic Techniques and Semidefinite Optimization</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.7230J" title="6.7230[J]" class="bubblelink code" onclick="return showCourse(this, '6.7230J');">6.7230[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.7210" title="6.7210[J]" class="bubblelink code" onclick="return showCourse(this, '6.7210');">6.7210[J]</a> or 15.093 </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=6.7230J" title="6.7230[J]" class="bubblelink code" onclick="return showCourse(this, '6.7230J');">6.7230[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>P. Parrilo</i></span></p> </div><h2 class="sectionhead">Logic</h2><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.504 Seminar in Logic</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.510" title="18.510" class="bubblelink code" onclick="return showCourse(this, '18.510');">18.510</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Students present and discuss the subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>H. Cohn</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.510 Introduction to Mathematical Logic and Set Theory</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Propositional and predicate logic. Zermelo-Fraenkel set theory. Ordinals and cardinals. Axiom of choice and transfinite induction. Elementary model theory: completeness, compactness, and Lowenheim-Skolem theorems. Godel's incompleteness theorem. </p> <p class="courseblockinstructors seemore"><span><i>H. Cohn</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.515 Mathematical Logic</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Spring)<br/> Not offered regularly; consult department</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">More rigorous treatment of basic mathematical logic, Godel's theorems, and Zermelo-Fraenkel set theory. First-order logic. Models and satisfaction. Deduction and proof. Soundness and completeness. Compactness and its consequences. Quantifier elimination. Recursive sets and functions. Incompleteness and undecidability. Ordinals and cardinals. Set-theoretic formalization of mathematics. </p> <p class="courseblockinstructors seemore"><span><i>Information: B. Poonen</i></span></p> </div><h2 class="sectionhead">Probability and Statistics</h2><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.600 Probability and Random Variables</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">4-0-8 units. REST</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=6.3700" title="6.3700" class="bubblelink code" onclick="return showCourse(this, '6.3700');">6.3700</a>, <a href="/search/?P=6.3702" title="6.3702" class="bubblelink code" onclick="return showCourse(this, '6.3702');">6.3702</a></span> </p> <p class="courseblockdesc">Probability spaces, random variables, distribution functions. Binomial, geometric, hypergeometric, Poisson distributions. Uniform, exponential, normal, gamma and beta distributions. Conditional probability, Bayes theorem, joint distributions. Chebyshev inequality, law of large numbers, and central limit theorem. Credit cannot also be received for 6.041A or 6.041B. </p> <p class="courseblockinstructors seemore"><span><i>Fall: S. Sheffield. Spring: J. Kelner</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.604 Seminar In Probability Theory (New)</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.05" title="18.05" class="bubblelink code" onclick="return showCourse(this, '18.05');">18.05</a> or <a href="/search/?P=18.600" title="18.600" class="bubblelink code" onclick="return showCourse(this, '18.600');">18.600</a> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Students work on group presentations on topics selected by students from a provided list of suggestions. Topics may include Benford's law, random walks and electrical networks, and Brownian motions. Assignments include three group presentations, two individual presentations, and a final individual term paper. Instruction in oral and written communication provided to effectively communicate about probability theory. Limited to 16.</p> <p class="courseblockinstructors seemore"><span><i>H. Cohn, S. Sheffield</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.615 Introduction to Stochastic Processes</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=6.3700" title="6.3700" class="bubblelink code" onclick="return showCourse(this, '6.3700');">6.3700</a> or <a href="/search/?P=18.600" title="18.600" class="bubblelink code" onclick="return showCourse(this, '18.600');">18.600</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Basics of stochastic processes. Markov chains, Poisson processes, random walks, birth and death processes, Brownian motion. </p> <p class="courseblockinstructors seemore"><span><i>J. He</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.619[J] Discrete Probability and Stochastic Processes</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=6.7720J" title="6.7720[J]" class="bubblelink code" onclick="return showCourse(this, '6.7720J');">6.7720[J]</a>, <a href="/search/?P=15.070J" title="15.070[J]" class="bubblelink code" onclick="return showCourse(this, '15.070J');">15.070[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.3702" title="6.3702" class="bubblelink code" onclick="return showCourse(this, '6.3702');">6.3702</a>, <a href="/search/?P=6.7700" title="6.7700[J]" class="bubblelink code" onclick="return showCourse(this, '6.7700');">6.7700[J]</a>, <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=15.070J" title="15.070[J]" class="bubblelink code" onclick="return showCourse(this, '15.070J');">15.070[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>G. Bresler, D. Gamarnik, E. Mossel,聽Y. Polyanskiy</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.642 Topics in Mathematics with Applications in Finance</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a>, <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, and (<a href="/search/?P=18.05" title="18.05" class="bubblelink code" onclick="return showCourse(this, '18.05');">18.05</a> or <a href="/search/?P=18.600" title="18.600" class="bubblelink code" onclick="return showCourse(this, '18.600');">18.600</a>) </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to mathematical concepts and techniques used in finance. Lectures focusing on linear algebra, probability, statistics, stochastic processes, and numerical methods are interspersed with lectures by financial sector professionals illustrating the corresponding application in the industry. Prior knowledge of economics or finance helpful but not required. </p> <p class="courseblockinstructors seemore"><span><i>P. Kempthorne, V. Strela, J. Xia</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.650[J] Fundamentals of Statistics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=IDS.014J" title="IDS.014[J]" class="bubblelink code" onclick="return showCourse(this, 'IDS.014J');">IDS.014[J]</a><br/> Subject meets with <a href="/search/?P=18.6501" title="18.6501" class="bubblelink code" onclick="return showCourse(this, '18.6501');">18.6501</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.3700" title="6.3700" class="bubblelink code" onclick="return showCourse(this, '6.3700');">6.3700</a> or <a href="/search/?P=18.600" title="18.600" class="bubblelink code" onclick="return showCourse(this, '18.600');">18.600</a> </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">4-0-8 units</span> </p> <p class="courseblockdesc">A rapid introduction to the theoretical foundations of statistical methods that are useful in many applications. Covers a broad range of topics in a short amount of time with the goal of providing a rigorous and cohesive understanding of the modern statistical landscape. Mathematical language is used for intuition and basic derivations but not proofs. Main topics include: parametric estimation, confidence intervals, hypothesis testing, Bayesian inference, and linear and logistic regression. Additional topics may include: causal inference, nonparametric estimation, and classification. </p> <p class="courseblockinstructors seemore"><span><i>Fall: P. Rigollet. Spring: A. Katsevich</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.6501 Fundamentals of Statistics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.650J" title="18.650[J]" class="bubblelink code" onclick="return showCourse(this, '18.650J');">18.650[J]</a>, <a href="/search/?P=IDS.014J" title="IDS.014[J]" class="bubblelink code" onclick="return showCourse(this, 'IDS.014J');">IDS.014[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.3700" title="6.3700" class="bubblelink code" onclick="return showCourse(this, '6.3700');">6.3700</a> or <a href="/search/?P=18.600" title="18.600" class="bubblelink code" onclick="return showCourse(this, '18.600');">18.600</a> </span><br/><span class="courseblockterms">G (Fall, Spring)</span><br/><span class="courseblockhours">4-0-8 units</span> </p> <p class="courseblockdesc">A rapid introduction to the theoretical foundations of statistical methods that are useful in many applications. Covers a broad range of topics in a short amount of time with the goal of providing a rigorous and cohesive understanding of the modern statistical landscape. Mathematical language is used for intuition and basic derivations but not proofs. Main topics include:聽parametric estimation, confidence intervals, hypothesis testing,聽Bayesian inference, and linear and logistic聽regression. Additional topics may include: causal inference,聽nonparametric聽estimation, and聽classification. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.650" title="18.650[J]" class="bubblelink code" onclick="return showCourse(this, '18.650');">18.650[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>Fall: P. Rigollet. Spring: A. Katsevich</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.655 Mathematical Statistics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: (<a href="/search/?P=18.650" title="18.650[J]" class="bubblelink code" onclick="return showCourse(this, '18.650');">18.650[J]</a> and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>)) or permission of instructor </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis.聽Prior exposure to both probability and statistics at the university level is assumed. </p> <p class="courseblockinstructors seemore"><span><i>P. Kempthorne</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.656[J] Mathematical Statistics: a Non-Asymptotic Approach</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=9.521J" title="9.521[J]" class="bubblelink code" onclick="return showCourse(this, '9.521J');">9.521[J]</a>, <a href="/search/?P=IDS.160J" title="IDS.160[J]" class="bubblelink code" onclick="return showCourse(this, 'IDS.160J');">IDS.160[J]</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=6.7700" title="6.7700[J]" class="bubblelink code" onclick="return showCourse(this, '6.7700');">6.7700[J]</a>, <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, and <a href="/search/?P=18.6501" title="18.6501" class="bubblelink code" onclick="return showCourse(this, '18.6501');">18.6501</a>) or permission of instructor </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=9.521J" title="9.521[J]" class="bubblelink code" onclick="return showCourse(this, '9.521J');">9.521[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>S. Rakhlin, P. Rigollet</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.657 Topics in Statistics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from term to term. </p> <p class="courseblockinstructors seemore"><span><i>P. Rigollet</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.675 Theory of Probability</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a> </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales. Prior exposure to probability (e.g., <a href="/search/?P=18.600" title="18.600" class="bubblelink code" onclick="return showCourse(this, '18.600');">18.600</a>) recommended. </p> <p class="courseblockinstructors seemore"><span><i>Y. Shenfeld</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.676 Stochastic Calculus</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.675" title="18.675" class="bubblelink code" onclick="return showCourse(this, '18.675');">18.675</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to stochastic processes, building on the fundamental example of Brownian motion. Topics include Brownian motion, continuous parameter martingales, Ito's theory of stochastic differential equations, Markov processes and partial differential equations, and may also include local time and excursion theory. Students should have familiarity with Lebesgue integration and its application to probability. </p> <p class="courseblockinstructors seemore"><span><i>N. Sun</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.677 Topics in Stochastic Processes</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.675" title="18.675" class="bubblelink code" onclick="return showCourse(this, '18.675');">18.675</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>A. Borodin</i></span></p> </div><h2 class="sectionhead">Algebra and Number Theory</h2><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.700 Linear Algebra</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units. REST</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=6.C06" title="6.C06[J]" class="bubblelink code" onclick="return showCourse(this, '6.C06');">6.C06[J]</a>, <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.C06" title="18.C06[J]" class="bubblelink code" onclick="return showCourse(this, '18.C06');">18.C06[J]</a>, <a href="/search/?P=ES.1806" title="ES.1806" class="bubblelink code" onclick="return showCourse(this, 'ES.1806');">ES.1806</a></span> </p> <p class="courseblockdesc">Vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. More emphasis on theory and proofs than in <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>. </p> <p class="courseblockinstructors seemore"><span><i>V. Kac</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.701 Algebra I</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>, <a href="/search/?P=18.090" title="18.090" class="bubblelink code" onclick="return showCourse(this, '18.090');">18.090</a>, or permission of instructor </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc"><a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>-<a href="/search/?P=18.702" title="18.702" class="bubblelink code" onclick="return showCourse(this, '18.702');">18.702</a> is more extensive and theoretical than the <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>-<a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a> sequence. Experience with proofs necessary. <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> focuses on group theory, geometry, and linear algebra. </p> <p class="courseblockinstructors seemore"><span><i>H. Cohn</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.702 Algebra II</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Continuation of <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>. Focuses on group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory. </p> <p class="courseblockinstructors seemore"><span><i>A. Negut</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.703 Modern Algebra</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.02|18.02A|18.022|18.024" onclick="return showCourse(this, '18.02|18.02A|18.022|18.024');">Calculus II (GIR)</a> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics: group theory, emphasizing finite groups; ring theory, including ideals and unique factorization in polynomial and Euclidean rings; field theory, including properties and applications of finite fields. <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a> and <a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a> together form a standard algebra sequence. </p> <p class="courseblockinstructors seemore"><span><i>V. G. Kac</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.704 Seminar in Algebra</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>, (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a> and <a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a>), or (<a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a> and <a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a>) </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Some experience with proofs required. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>K. Vashaw</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.705 Commutative Algebra</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.702" title="18.702" class="bubblelink code" onclick="return showCourse(this, '18.702');">18.702</a> </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Exactness, direct limits, tensor products, Cayley-Hamilton theorem, integral dependence, localization, Cohen-Seidenberg theory, Noether normalization, Nullstellensatz, chain conditions, primary decomposition, length, Hilbert functions, dimension theory, completion, Dedekind domains. </p> <p class="courseblockinstructors seemore"><span><i>W. Zhang</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.706 Noncommutative Algebra</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.702" title="18.702" class="bubblelink code" onclick="return showCourse(this, '18.702');">18.702</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Topics may include Wedderburn theory and structure of Artinian rings, Morita equivalence and elements of category theory, localization and Goldie's theorem, central simple algebras and the Brauer group, representations, polynomial identity rings, invariant theory growth of algebras, Gelfand-Kirillov dimension. </p> <p class="courseblockinstructors seemore"><span><i>R. Bezrukavnikov</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.708 Topics in Algebra</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.705" title="18.705" class="bubblelink code" onclick="return showCourse(this, '18.705');">18.705</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>P. I. Etingof</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.715 Introduction to Representation Theory</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.702" title="18.702" class="bubblelink code" onclick="return showCourse(this, '18.702');">18.702</a> or <a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Algebras, representations, Schur's lemma. Representations of SL(2). Representations of finite groups, Maschke's theorem, characters, applications. Induced representations, Burnside's theorem, Mackey formula, Frobenius reciprocity. Representations of quivers. </p> <p class="courseblockinstructors seemore"><span><i>G. Lusztig</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.721 Introduction to Algebraic Geometry</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.702" title="18.702" class="bubblelink code" onclick="return showCourse(this, '18.702');">18.702</a> and <a href="/search/?P=18.901" title="18.901" class="bubblelink code" onclick="return showCourse(this, '18.901');">18.901</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Presents basic examples of complex algebraic varieties, affine and projective algebraic geometry, sheaves, cohomology. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.725 Algebraic Geometry I</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None. <i>Coreq: <a href="/search/?P=18.705" title="18.705" class="bubblelink code" onclick="return showCourse(this, '18.705');">18.705</a></i> </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduces the basic notions and techniques of modern algebraic geometry. Covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. Introduction to the language of schemes and properties of morphisms. Knowledge of elementary algebraic topology, elementary differential geometry recommended, but not required. </p> <p class="courseblockinstructors seemore"><span><i>D. Maulik</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.726 Algebraic Geometry II</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.725" title="18.725" class="bubblelink code" onclick="return showCourse(this, '18.725');">18.725</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Continuation of the introduction to algebraic geometry given in <a href="/search/?P=18.725" title="18.725" class="bubblelink code" onclick="return showCourse(this, '18.725');">18.725</a>. More advanced properties of the varieties and morphisms of schemes, as well as sheaf cohomology. </p> <p class="courseblockinstructors seemore"><span><i>D. Maulik</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.727 Topics in Algebraic Geometry</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.725" title="18.725" class="bubblelink code" onclick="return showCourse(this, '18.725');">18.725</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>A. Negut</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.737 Algebraic Groups</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.705" title="18.705" class="bubblelink code" onclick="return showCourse(this, '18.705');">18.705</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Structure of linear algebraic groups over an algebraically closed field, with emphasis on reductive groups. Representations of groups over a finite field using methods from etale cohomology. Some results from algebraic geometry are stated without proof. </p> <p class="courseblockinstructors seemore"><span><i>J.-L. Kim</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.745 Lie Groups and Lie Algebras I</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: (<a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> or <a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Covers fundamentals of the theory of Lie algebras and related groups. Topics may include theorems of Engel and Lie; enveloping algebra, Poincare-Birkhoff-Witt theorem; classification and construction of semisimple Lie algebras; the center of their enveloping algebras; elements of representation theory; compact Lie groups and/or finite Chevalley groups. </p> <p class="courseblockinstructors seemore"><span><i>V. G. Kac</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.747 Infinite-dimensional Lie Algebras</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.745" title="18.745" class="bubblelink code" onclick="return showCourse(this, '18.745');">18.745</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>P. I. Etingof</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.748 Topics in Lie Theory</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>P. I. Etingof</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.755 Lie Groups and Lie Algebras II</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.745" title="18.745" class="bubblelink code" onclick="return showCourse(this, '18.745');">18.745</a> or permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">A more in-depth treatment of Lie groups and Lie algebras. Topics may include homogeneous spaces and groups of automorphisms; representations of compact groups and their geometric realizations, Peter-Weyl theorem; invariant differential forms and cohomology of Lie groups and homogeneous spaces; complex reductive Lie groups, classification of real reductive groups. </p> <p class="courseblockinstructors seemore"><span><i>Z. Yun</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.757 Representations of Lie Groups</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.745" title="18.745" class="bubblelink code" onclick="return showCourse(this, '18.745');">18.745</a> or <a href="/search/?P=18.755" title="18.755" class="bubblelink code" onclick="return showCourse(this, '18.755');">18.755</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Covers representations of locally compact groups, with emphasis on compact groups and abelian groups. Includes Peter-Weyl theorem and Cartan-Weyl highest weight theory for compact Lie groups. </p> <p class="courseblockinstructors seemore"><span><i>P. I. Etingof</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.781 Theory of Numbers</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">An elementary introduction to number theory with no algebraic prerequisites. Primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, partitions. </p> <p class="courseblockinstructors seemore"><span><i>M.-T. Trinh</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.782 Introduction to Arithmetic Geometry</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.702" title="18.702" class="bubblelink code" onclick="return showCourse(this, '18.702');">18.702</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: U (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Exposes students to arithmetic geometry, motivated by the problem of finding rational points on curves. Includes an introduction to p-adic numbers and some fundamental results from number theory and algebraic geometry, such as the Hasse-Minkowski theorem and the Riemann-Roch theorem for curves. Additional topics may include Mordell's theorem, the Weil conjectures, and Jacobian varieties. </p> <p class="courseblockinstructors seemore"><span><i>S. Chidambaram</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.783 Elliptic Curves</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.7831" title="18.7831" class="bubblelink code" onclick="return showCourse(this, '18.7831');">18.7831</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.702" title="18.702" class="bubblelink code" onclick="return showCourse(this, '18.702');">18.702</a>, <a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a>, or permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem. </p> <p class="courseblockinstructors seemore"><span><i>A. Sutherland</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.7831 Elliptic Curves</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.783" title="18.783" class="bubblelink code" onclick="return showCourse(this, '18.783');">18.783</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.702" title="18.702" class="bubblelink code" onclick="return showCourse(this, '18.702');">18.702</a>, <a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a>, or permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.783" title="18.783" class="bubblelink code" onclick="return showCourse(this, '18.783');">18.783</a>. </p> <p class="courseblockinstructors seemore"><span><i>A. Sutherland</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.784 Seminar in Number Theory</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> or (<a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a> and (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a> or <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>)) </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>A. Landesman</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.785 Number Theory I</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None. <i>Coreq: <a href="/search/?P=18.705" title="18.705" class="bubblelink code" onclick="return showCourse(this, '18.705');">18.705</a></i> </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Dedekind domains, unique factorization of ideals, splitting of primes. Lattice methods, finiteness of the class group, Dirichlet's unit theorem. Local fields, ramification, discriminants. Zeta and L-functions, analytic class number formula. Adeles and ideles. Statements of class field theory and the Chebotarev density theorem. </p> <p class="courseblockinstructors seemore"><span><i>B. Poonen</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.786 Number Theory II</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.785" title="18.785" class="bubblelink code" onclick="return showCourse(this, '18.785');">18.785</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Continuation of <a href="/search/?P=18.785" title="18.785" class="bubblelink code" onclick="return showCourse(this, '18.785');">18.785</a>. More advanced topics in number theory, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, or quadratic forms. </p> <p class="courseblockinstructors seemore"><span><i>A. Sutherland</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.787 Topics in Number Theory</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Topics vary from year to year. </p> <p class="courseblockinstructors seemore"><span><i>W. Zhang</i></span></p> </div><h2 class="sectionhead">Mathematics Laboratory</h2><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.821 Project Laboratory in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Two mathematics subjects numbered 18.100 or above </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">3-6-3 units. Institute LAB</span> </p> <p class="courseblockdesc">Guided research in mathematics, employing the scientific method. Students confront puzzling and complex mathematical situations, through the acquisition of data by computer, pencil and paper, or physical experimentation, and attempt to explain them mathematically. Students choose three projects from a large collection of options. Each project results in a laboratory report subject to revision; oral presentation on one or two projects. Projects drawn from many areas, including dynamical systems, number theory, algebra, fluid mechanics, asymptotic analysis, knot theory, and probability. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>Fall: A. Negut. Spring: L. Piccirillo</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.896[J] Leadership and Professional Strategies & Skills Training (LEAPS), Part I: Advancing Your Professional Strategies and Skills</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=5.961J" title="5.961[J]" class="bubblelink code" onclick="return showCourse(this, '5.961J');">5.961[J]</a>, <a href="/search/?P=8.396J" title="8.396[J]" class="bubblelink code" onclick="return showCourse(this, '8.396J');">8.396[J]</a>, <a href="/search/?P=9.980J" title="9.980[J]" class="bubblelink code" onclick="return showCourse(this, '9.980J');">9.980[J]</a>, <a href="/search/?P=12.396J" title="12.396[J]" class="bubblelink code" onclick="return showCourse(this, '12.396J');">12.396[J]</a></span><br/><span class="courseblockprereq">Prereq: None </span><br/><span class="courseblockterms">G (Spring; second half of term)</span><br/><span class="courseblockhours">2-0-1 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=8.396J" title="8.396[J]" class="bubblelink code" onclick="return showCourse(this, '8.396J');">8.396[J]</a>. Limited to 80.</p> <p class="courseblockinstructors seemore"><span><i>A. Frebel</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.897[J] Leadership and Professional Strategies & Skills Training (LEAPS), Part II: Developing Your Leadership Competencies</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=5.962J" title="5.962[J]" class="bubblelink code" onclick="return showCourse(this, '5.962J');">5.962[J]</a>, <a href="/search/?P=8.397J" title="8.397[J]" class="bubblelink code" onclick="return showCourse(this, '8.397J');">8.397[J]</a>, <a href="/search/?P=9.981J" title="9.981[J]" class="bubblelink code" onclick="return showCourse(this, '9.981J');">9.981[J]</a>, <a href="/search/?P=12.397J" title="12.397[J]" class="bubblelink code" onclick="return showCourse(this, '12.397J');">12.397[J]</a></span><br/><span class="courseblockprereq">Prereq: None </span><br/><span class="courseblockterms">G (Spring; first half of term)</span><br/><span class="courseblockhours">2-0-1 units</span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=8.397J" title="8.397[J]" class="bubblelink code" onclick="return showCourse(this, '8.397J');">8.397[J]</a>. Limited to 80.</p> <p class="courseblockinstructors seemore"><span><i>D. Rigos</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.899 Internship in Mathematics (New)</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None </span><br/><span class="courseblockterms">G (Fall, Spring, Summer)</span><br/><span class="courseblockhours">Units arranged [P/D/F]</span> </p> <p class="courseblockdesc">Provides academic credit for students pursuing internships to gain practical experience applications of mathematical concepts and methods as related to their field of research. </p> <p class="courseblockinstructors seemore"><span><i>J. Kelner, D. Maulik, Z. Yun</i></span></p> </div><h2 class="sectionhead">Topology and Geometry</h2><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.900 Geometry and Topology in the Plane</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a> or <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to selected aspects of geometry and topology, using concepts that can be visualized easily. Mixes geometric topics (such as hyperbolic geometry or billiards) and more topological ones (such as loops in the plane). Suitable for students with no prior exposure to differential geometry or topology. </p> <p class="courseblockinstructors seemore"><span><i>P. Seidel</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.901 Introduction to Topology</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.9011" title="18.9011" class="bubblelink code" onclick="return showCourse(this, '18.9011');">18.9011</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>, or permission of instructor </span><br/><span class="courseblockterms">U (Fall, Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, and the fundamental group. </p> <p class="courseblockinstructors seemore"><span><i>Fall: A. Pieloch. Spring: R. Jiang</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.9011 Introduction to Topology</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.901" title="18.901" class="bubblelink code" onclick="return showCourse(this, '18.901');">18.901</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>, or permission of instructor </span><br/><span class="courseblockterms">G (Fall, Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, and the fundamental group. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.901" title="18.901" class="bubblelink code" onclick="return showCourse(this, '18.901');">18.901</a>. </p> <p class="courseblockinstructors seemore"><span><i>Fall: A. Pieloch. Spring: R. Jiang</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.904 Seminar in Topology</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.901" title="18.901" class="bubblelink code" onclick="return showCourse(this, '18.901');">18.901</a> </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>A. Ward</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.905 Algebraic Topology I</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.901" title="18.901" class="bubblelink code" onclick="return showCourse(this, '18.901');">18.901</a> and (<a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a> or <a href="/search/?P=18.703" title="18.703" class="bubblelink code" onclick="return showCourse(this, '18.703');">18.703</a>) </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Singular homology, CW complexes, universal coefficient and K眉nneth theorems, cohomology, cup products, Poincar茅 duality. </p> <p class="courseblockinstructors seemore"><span><i>D. Alvarez-Gavela</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.906 Algebraic Topology II</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.905" title="18.905" class="bubblelink code" onclick="return showCourse(this, '18.905');">18.905</a> and (<a href="/search/?P=18.101" title="18.101" class="bubblelink code" onclick="return showCourse(this, '18.101');">18.101</a> or <a href="/search/?P=18.965" title="18.965" class="bubblelink code" onclick="return showCourse(this, '18.965');">18.965</a>) </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Continues the introduction to Algebraic Topology from <a href="/search/?P=18.905" title="18.905" class="bubblelink code" onclick="return showCourse(this, '18.905');">18.905</a>. Topics include basic homotopy theory, spectral sequences, characteristic classes, and cohomology operations. </p> <p class="courseblockinstructors seemore"><span><i>T. S. Mrowka</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.917 Topics in Algebraic Topology</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.906" title="18.906" class="bubblelink code" onclick="return showCourse(this, '18.906');">18.906</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Content varies from year to year. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas. </p> <p class="courseblockinstructors seemore"><span><i>Information: T. Schlank</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.919 Graduate Topology Seminar</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.906" title="18.906" class="bubblelink code" onclick="return showCourse(this, '18.906');">18.906</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Study and discussion of important original papers in the various parts of topology. Open to all students who have taken <a href="/search/?P=18.906" title="18.906" class="bubblelink code" onclick="return showCourse(this, '18.906');">18.906</a> or the equivalent, not only prospective topologists. </p> <p class="courseblockinstructors seemore"><span><i>T. Schlank</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.937 Topics in Geometric Topology</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Fall, Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Content varies from year to year. Introduces new and significant developments in geometric topology. </p> <p class="courseblockinstructors seemore"><span><i>T. S. Mrowka</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.950 Differential Geometry</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.9501" title="18.9501" class="bubblelink code" onclick="return showCourse(this, '18.9501');">18.9501</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to differential geometry, centered on notions of curvature. Starts with curves in the plane, and proceeds to higher dimensional submanifolds. Computations in coordinate charts: first and second fundamental form, Christoffel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Examples such as hyperbolic space. </p> <p class="courseblockinstructors seemore"><span><i>G. Franz</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.9501 Differential Geometry</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Subject meets with <a href="/search/?P=18.950" title="18.950" class="bubblelink code" onclick="return showCourse(this, '18.950');">18.950</a></span><br/><span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Introduction to differential geometry, centered on notions of curvature. Starts with curves in the plane, and proceeds to higher dimensional submanifolds. Computations in coordinate charts: first and second fundamental form, Christoffel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Examples such as hyperbolic space. Students in Course 18 must register for the undergraduate version, <a href="/search/?P=18.950" title="18.950" class="bubblelink code" onclick="return showCourse(this, '18.950');">18.950</a>. </p> <p class="courseblockinstructors seemore"><span><i>G. Franz</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.952 Theory of Differential Forms</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.101" title="18.101" class="bubblelink code" onclick="return showCourse(this, '18.101');">18.101</a> and (<a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a> or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) </span><br/><span class="courseblockterms">U (Spring)<br/> Not offered regularly; consult department</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Multilinear algebra: tensors and exterior forms. Differential forms on R<sup>n</sup>: exterior differentiation, the pull-back operation and the Poincar茅 lemma. Applications to physics: Maxwell's equations from the differential form perspective. Integration of forms on open sets of R<sup>n</sup>. The change of variables formula revisited. The degree of a differentiable mapping. Differential forms on manifolds and De Rham theory. Integration of forms on manifolds and Stokes' theorem. The push-forward operation for forms. Thom forms and intersection theory. Applications to differential topology. </p> <p class="courseblockinstructors seemore"><span><i>V. W. Guillemin</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.965 Geometry of Manifolds I</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.101" title="18.101" class="bubblelink code" onclick="return showCourse(this, '18.101');">18.101</a>, <a href="/search/?P=18.950" title="18.950" class="bubblelink code" onclick="return showCourse(this, '18.950');">18.950</a>, or <a href="/search/?P=18.952" title="18.952" class="bubblelink code" onclick="return showCourse(this, '18.952');">18.952</a> </span><br/><span class="courseblockterms">G (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Differential forms, introduction to Lie groups, the DeRham theorem, Riemannian manifolds, curvature, the Hodge theory. <a href="/search/?P=18.966" title="18.966" class="bubblelink code" onclick="return showCourse(this, '18.966');">18.966</a> is a continuation of <a href="/search/?P=18.965" title="18.965" class="bubblelink code" onclick="return showCourse(this, '18.965');">18.965</a> and focuses more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology. Prior exposure to calculus on manifolds, as in <a href="/search/?P=18.952" title="18.952" class="bubblelink code" onclick="return showCourse(this, '18.952');">18.952</a>, recommended. </p> <p class="courseblockinstructors seemore"><span><i>W. Minicozzi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.966 Geometry of Manifolds II</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.965" title="18.965" class="bubblelink code" onclick="return showCourse(this, '18.965');">18.965</a> </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Continuation of <a href="/search/?P=18.965" title="18.965" class="bubblelink code" onclick="return showCourse(this, '18.965');">18.965</a>, focusing more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology. </p> <p class="courseblockinstructors seemore"><span><i>T. Colding</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.968 Topics in Geometry</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: <a href="/search/?P=18.965" title="18.965" class="bubblelink code" onclick="return showCourse(this, '18.965');">18.965</a> </span><br/><span class="courseblockterms">Acad Year 2024-2025: G (Spring)<br/> Acad Year 2025-2026: Not offered</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Content varies from year to year. </p> <p class="courseblockinstructors seemore"><span><i>P. Seidel</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.979 Graduate Geometry Seminar</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Spring)<br/> Not offered regularly; consult department</span><br/><span class="courseblockhours">3-0-9 units</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Content varies from year to year. Study of classical papers in geometry and in applications of analysis to geometry and topology. </p> <p class="courseblockinstructors seemore"><span><i>T. Mrowka</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.994 Seminar in Geometry</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: (<a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a>, <a href="/search/?P=18.700" title="18.700" class="bubblelink code" onclick="return showCourse(this, '18.700');">18.700</a>, or <a href="/search/?P=18.701" title="18.701" class="bubblelink code" onclick="return showCourse(this, '18.701');">18.701</a>) and (<a href="/search/?P=18.100A" title="18.100A" class="bubblelink code" onclick="return showCourse(this, '18.100A');">18.100A</a>, <a href="/search/?P=18.100B" title="18.100B" class="bubblelink code" onclick="return showCourse(this, '18.100B');">18.100B</a>, <a href="/search/?P=18.100P" title="18.100P" class="bubblelink code" onclick="return showCourse(this, '18.100P');">18.100P</a>, or <a href="/search/?P=18.100Q" title="18.100Q" class="bubblelink code" onclick="return showCourse(this, '18.100Q');">18.100Q</a>) </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Students present and discuss subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.</p> <p class="courseblockinstructors seemore"><span><i>Q. Deng</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.999 Research in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Fall, IAP, Spring, Summer)</span><br/><span class="courseblockhours">Units arranged</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Opportunity for study of graduate-level topics in mathematics under the supervision of a member of the department. For graduate students desiring advanced work not provided in regular subjects. </p> <p class="courseblockinstructors seemore"><span><i>Information: W. Minicozzi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.C20[J] Introduction to Computational Science and Engineering</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=9.C20J" title="9.C20[J]" class="bubblelink code" onclick="return showCourse(this, '9.C20J');">9.C20[J]</a>, <a href="/search/?P=16.C20J" title="16.C20[J]" class="bubblelink code" onclick="return showCourse(this, '16.C20J');">16.C20[J]</a>, <a href="/search/?P=CSE.C20J" title="CSE.C20[J]" class="bubblelink code" onclick="return showCourse(this, 'CSE.C20J');">CSE.C20[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.100A" title="6.100A" class="bubblelink code" onclick="return showCourse(this, '6.100A');">6.100A</a>; <i>Coreq: <a href="/search/?P=8.01" title="8.01" class="bubblelink code" onclick="return showCourse(this, '8.01');">8.01</a> and <a href="/search/?P=18.01" title="18.01" class="bubblelink code" onclick="return showCourse(this, '18.01');">18.01</a></i> </span><br/><span class="courseblockterms">U (Fall, Spring; second half of term)</span><br/><span class="courseblockhours">2-0-4 units</span><br/><span class="courseblockoptional">Credit cannot also be received for <a href="/search/?P=6.100B" title="6.100B" class="bubblelink code" onclick="return showCourse(this, '6.100B');">6.100B</a></span> </p> <p class="courseblockdesc">See description under subject <a href="/search/?P=16.C20J" title="16.C20[J]" class="bubblelink code" onclick="return showCourse(this, '16.C20J');">16.C20[J]</a>. </p> <p class="courseblockinstructors seemore"><span><i>D. L. Darmofal, N. Seethapathi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.C25[J] Real World Computation with Julia</strong></span></h4> <p class="courseblockextra"> <span class="courseblockcluster">Same subject as <a href="/search/?P=1.C25J" title="1.C25[J]" class="bubblelink code" onclick="return showCourse(this, '1.C25J');">1.C25[J]</a>, <a href="/search/?P=6.C25J" title="6.C25[J]" class="bubblelink code" onclick="return showCourse(this, '6.C25J');">6.C25[J]</a>, <a href="/search/?P=12.C25J" title="12.C25[J]" class="bubblelink code" onclick="return showCourse(this, '12.C25J');">12.C25[J]</a>, <a href="/search/?P=16.C25J" title="16.C25[J]" class="bubblelink code" onclick="return showCourse(this, '16.C25J');">16.C25[J]</a>, <a href="/search/?P=22.C25J" title="22.C25[J]" class="bubblelink code" onclick="return showCourse(this, '22.C25J');">22.C25[J]</a></span><br/><span class="courseblockprereq">Prereq: <a href="/search/?P=6.100A" title="6.100A" class="bubblelink code" onclick="return showCourse(this, '6.100A');">6.100A</a>, <a href="/search/?P=18.03" title="18.03" class="bubblelink code" onclick="return showCourse(this, '18.03');">18.03</a>, and <a href="/search/?P=18.06" title="18.06" class="bubblelink code" onclick="return showCourse(this, '18.06');">18.06</a> </span><br/><span class="courseblockterms">U (Fall)</span><br/><span class="courseblockhours">3-0-9 units</span> </p> <p class="courseblockdesc">Focuses on algorithms and techniques for writing and using modern technical software in a job, lab, or research group environment that may consist of interdisciplinary teams, where performance may be critical, and where the software needs to be flexible and adaptable. Topics include automatic differentiation, matrix calculus, scientific machine learning, parallel and GPU computing, and performance optimization with introductory applications to climate science, economics, agent-based modeling, and other areas. Labs and projects focus on performant, readable, composable algorithms, and software. Programming will be in Julia. Expects students to have some familiarity with Python, Matlab, or R. No Julia experience necessary. </p> <p class="courseblockinstructors seemore"><span><i>A. Edelman, R. Ferrari, B. Forget, C. Leiseron,Y. Marzouk, J. Williams</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.UR Undergraduate Research</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">U (Fall, IAP, Spring, Summer)</span><br/><span class="courseblockhours">Units arranged [P/D/F]</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Undergraduate research opportunities in mathematics. Permission required in advance to register for this subject. For further information, consult the departmental coordinator. </p> <p class="courseblockinstructors seemore"><span><i>Information: W. Minicozzi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.TAC Classroom Teaching in Mathematics (New)</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: None </span><br/><span class="courseblockterms">G (Fall, IAP, Spring)</span><br/><span class="courseblockhours">12-0-0 units</span> </p> <p class="courseblockdesc">For classroom training in Mathematics, in cases where teaching assignment is to fulfill academic teaching requirement by the department. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.THG Graduate Thesis</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Fall, IAP, Spring, Summer)</span><br/><span class="courseblockhours">Units arranged</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Program of research leading to the writing of a Ph.D. thesis; to be arranged by the student and an appropriate MIT faculty member. </p> <p class="courseblockinstructors seemore"><span><i>Information: W. Minicozzi</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.S096 Special Subject in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">U (Spring)</span><br/><span class="courseblockhours">Units arranged</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.S097 Special Subject in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">U (IAP)</span><br/><span class="courseblockhours">Units arranged [P/D/F]</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. <a href="/search/?P=18.S097" title="18.S097" class="bubblelink code" onclick="return showCourse(this, '18.S097');">18.S097</a> is graded P/D/F. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.S190 Special Subject in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">U (IAP)</span><br/><span class="courseblockhours">Units arranged</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.S191 Special Subject in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">Acad Year 2024-2025: Not offered<br/> Acad Year 2025-2026: U (IAP)</span><br/><span class="courseblockhours">Units arranged</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.S995 Special Subject in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">Units arranged</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the mathematics faculty on an ad hoc basis, subject to departmental approval. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.S996 Special Subject in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">Units arranged</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.S997 Special Subject in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">Units arranged</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div><div class="courseblock"> <h4 class="courseblocktitle"><span font="Meta Pro"><strong>18.S998 Special Subject in Mathematics</strong></span></h4> <p class="courseblockextra"> <span class="courseblockprereq">Prereq: Permission of instructor </span><br/><span class="courseblockterms">G (Spring)</span><br/><span class="courseblockhours">Units arranged</span><br/><span class="courseblockoptional">Can be repeated for credit.</span> </p> <p class="courseblockdesc">Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. </p> <p class="courseblockinstructors seemore"><span><i>Staff</i></span></p> </div></div> </div> </div>  </div> </div>  <div id="footer" role="contentinfo"> <div class="wrap"> <div id="footer-links"> <a href="http://mitadmissions.org/apply" target="_blank">Admissions</a><a href="http://sfs.mit.edu/" target="_blank">Financial Aid</a><a href="http://web.mit.edu/registrar/" target="_blank">Registrar</a><a href="http://web.mit.edu/iap/" target="_blank">IAP</a><a href="/summer/">Summer</a><a href="http://professional.mit.edu" target="_blank">Professional Education</a><a href="https://www.edx.org/school/mitx" class="mitx" target="_blank">MITx</a><a href="https://outreach.mit.edu/" target="_blank">K-12</a><a href="http://whereis.mit.edu/" target="_blank">Campus Map</a><br/><a href="https://officesdirectory.mit.edu/" target="_blank">Directories</a><a href="/about-bulletin/">About the Bulletin</a><a href="/nondiscrimination-policy/">Nondiscrimination Policy</a><a href="/changelog/">Changes</a><a href="/help/">Help</a><a href="https://accessibility.mit.edu/" target="_blank">Accessibility</a> </div> <div id="footer-branding" class="clearfix"> <div class="foot-col" id="right"> <div id="footer-mit-logo"><img src="/images/mit-logo-footer.svg" alt="MIT" width="185px" /></div> </div> <div class="foot-col" id="left"> <div id="footer-bulletin"><img src="/images/mit-bulletin-logo-footer-24-25.png" alt="MIT Academic Bulletin" /></div> <div id="footer-contact">77 Massachusetts Avenue<br/>Cambridge, MA 02139-4307</div> </div> </div> </div> <a href="#header" id="totop"><span class="sr-only">Back to top</span></a> </div>  <div id="print-dialog" class="screen noscript" role="dialog"> <div class="print-header"> Print Options </div> <div class="print-body"> <div class="print-option"> <div class="option-head"> <a href="#" onclick="hidePrintDialog();window.print();return false">Send Page to Printer</a> </div> <p> <em>Print this page.</em> </p> </div> <div class="print-option"><div class="option-head"><a href="mathematics.pdf" target="_blank">Download PDF of this Page</a></div><p><em>The PDF includes all information on this page and its related tabs. 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