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geometric measure theory in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/3187/#Item_3" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="integration_theory">Integration theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/integration">integration</a></strong></p> <table><thead><tr><th>analytic integration</th><th>cohomological integration</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/measure">measure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/virtual+fundamental+class">virtual</a>) <a class="existingWikiWord" href="/nlab/show/fundamental+class">fundamental class</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Riemann+integration">Riemann</a>/<a class="existingWikiWord" href="/nlab/show/Lebesgue+integration">Lebesgue integration</a> <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">of differential forms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward in generalized cohomology</a>/<a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">in differential cohomology</a></td></tr> </tbody></table> <h2 id="analytic_integration">Analytic integration</h2> <p><a class="existingWikiWord" href="/nlab/show/integral+calculus">integral calculus</a></p> <p><a class="existingWikiWord" href="/nlab/show/Riemann+integration">Riemann integration</a>, <a class="existingWikiWord" href="/nlab/show/Lebesgue+integration">Lebesgue integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/line+integral">line integral</a>/<a class="existingWikiWord" href="/nlab/show/contour+integration">contour integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a>, <a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a>, <a class="existingWikiWord" href="/nlab/show/fermionic+path+integral">fermionic path integral</a></p> <p><a class="existingWikiWord" href="/nlab/show/Kontsevich+integral">Kontsevich integral</a>, <a class="existingWikiWord" href="/nlab/show/Selberg+integral">Selberg integral</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+Selberg+integral">elliptic Selberg integral</a></p> <h2 id="cohomological_integration">Cohomological integration</h2> <p><a class="existingWikiWord" href="/nlab/show/integration+in+ordinary+differential+cohomology">integration in ordinary differential cohomology</a></p> <p><a class="existingWikiWord" href="/nlab/show/integration+in+differential+K-theory">integration in differential K-theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/Riemann-Roch+theorem">Riemann-Roch theorem</a></p> <h2 id="variants">Variants</h2> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a></p> <p><a class="existingWikiWord" href="/nlab/show/Batalin-Vilkovisky+integral">Batalin-Vilkovisky integral</a></p></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Geometric measure theory</em> and <em>geometric integration theory</em> studies various <a class="existingWikiWord" href="/nlab/show/measures">measures</a> of subsets of <a class="existingWikiWord" href="/nlab/show/Euclidean+spaces">Euclidean spaces</a> and possibly of some geometric generalizations) and their geometric properties. Especially, one studies <a class="existingWikiWord" href="/nlab/show/rectifiability">rectifiability</a> of subsets of some lower dimensionality, to define notions like area, arc length etc. and to study <a class="existingWikiWord" href="/nlab/show/distributions">distributions</a> and <a class="existingWikiWord" href="/nlab/show/currents">currents</a> on such spaces. Very central questions and motivations belong to the <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational problems</a> including the study of <a class="existingWikiWord" href="/nlab/show/minimal+surface">minimal surfaces</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+integration">cohomological integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/current+%28distribution+theory%29">current (distribution theory)</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Herbert+Federer">Herbert Federer</a>, <em>Geometric measure theory</em>, Grundlehren <strong>153</strong>, Springer (1969, 1996) [<a href="https://doi.org/10.1007/978-3-642-62010-2">doi:10.1007/978-3-642-62010-2</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Victor+Guillemin">Victor Guillemin</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Sternberg">Shlomo Sternberg</a>, <em>Geometric asymptotics</em>, Mathematical Surveys and Monographs <strong>14</strong>, AMS (1977) [<a href="https://bookstore.ams.org/surv-14">ams:surv-14</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Jost">Jürgen Jost</a>, <em>The geometric calculus of variations: a short survey and a list of open problems</em>, Exposition. Math. <strong>6</strong> (1988), no. 2, 111–143, <a href="http://www.ams.org/mathscinet-getitem?mr=938179">MR89h:58036</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steven+G.+Krantz">Steven G. Krantz</a>, <a class="existingWikiWord" href="/nlab/show/Harold+Parks">Harold Parks</a>, <em>Geometric integration theory</em>, Springer (2008) [<a href="http://www.math.wustl.edu/~sk/books/root.pdf">pdf</a>]</p> </li> <li> <p>Frederick J., Jr. Almgren, Almgren’s big regularity paper (book form of a 1970s preprint)</p> </li> <li> <p>F. J. Almgren, Jr., <em>Geometric measure theory and elliptic variational problems</em>, Proc. ICM Nice 1970, vol. 2, <a href="http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0813.0820.ocr.djvu">djvu:350 K</a>, <a href="http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0813.0820.ocr.pdf">pdf:649 K</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frank+Morgan">Frank Morgan</a>, <em>Geometric measure theory: a beginner’s guide</em>, Academic Press (2016) [<a href="https://doi.org/10.1016/C2015-0-01918-9">doi:10.1016/C2015-0-01918-9</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eom">eom</a>: T.C.O’Neil (2001), <a href="http://eom.springer.de/G/g130040.htm">Geometric measure theory</a></p> </li> <li> <p>H. Federer, W. H. Fleming, <em>Normal and integral currents</em>, Ann. of Math. (2) <strong>72</strong> 1960 458–520, <a href="http://www.ams.org/mathscinet-getitem?mr=123260">MR123260</a>, <a href="http://dx.doi.org/10.2307/1970227">doi</a></p> </li> <li> <p>Stephen H. Schanuel, <em>What is the length of a potato? An introduction to geometric measure theory</em>, in: Categories in continuum physics (Buffalo, N.Y., 1982), 118–126, Lecture Notes in Math. <strong>1174</strong>, Springer 1986, <a href="http://www.ams.org/mathscinet-getitem?mr=842922">MR842922</a>,<a href="http://dx.doi.org/10.1007/BFb0076939">doi</a></p> </li> <li> <p>Rodolfo Rios-Zertuche, <em>The variational structure of the space of holonomic measures</em>, <a href="http://arxiv.org/abs/1408.5785">arxiv1408.5785</a></p> </li> </ul> <p>See also:</p> <ul> <li>wikipedia <a href="http://en.wikipedia.org/wiki/Geometric_measure_theory">geometric measure theory</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 10, 2023 at 07:37:03. 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