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Michael Hopkins in nLab
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width: 0.3em;"></span> <a 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/5753/#Item_2" 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></title></head> <body> <p>Michael Jerome Hopkins is a mathematician at Harvard University. He got his PhD from Northwestern University in 1984, advised by <a class="existingWikiWord" href="/nlab/show/Mark+Mahowald">Mark Mahowald</a>.</p> <ul> <li> <p><a href="http://www.harvardscience.harvard.edu/directory/researchers/michael-j-hopkins">website</a></p> </li> <li> <p><a href="http://en.wikipedia.org/wiki/Michael_J._Hopkins">Wikipedia entry</a></p> </li> </ul> <p>Hopkins is a world leading researcher in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> and (<a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>-)<a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p>Among his notable achievements are his work on the <a class="existingWikiWord" href="/nlab/show/Ravenel+conjectures">Ravenel conjectures</a>, the introduction and discussion of the <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> <em><a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></em> and its <a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation</a>, a formalization and construction of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>, the proof of the <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a> problem. More recently via <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>‘s work on the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> Hopkins participates in work related to the foundations of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>.</p> <h2 id="selected_writings">Selected writings</h2> <p>On <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a> (notes by <a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>), <em>Spectra and stable homotopy theory</em>, 2012 (<a href="http://math.uchicago.edu/~amathew/256y.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/HopkinsMathewStableHomotopyTheory.pdf" title="pdf">pdf</a>)</li> </ul> <p>Introducing generalized <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> motivated by the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> partition function:</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2Cand+M-Theory">Quadratic Functions in Geometry, Topology,and M-Theory</a></em></li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/topological+quantum+field+theory">topological quantum field theory</a>:</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/Topological+Quantum+Field+Theories+from+Compact+Lie+Groups">Topological Quantum Field Theories from Compact Lie Groups</a></em></li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/ambidextrous+adjunctions">ambidextrous adjunctions</a> in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/Ambidexterity+in+K%28n%29-Local+Stable+Homotopy+Theory">Ambidexterity in K(n)-Local Stable Homotopy Theory</a></em></li> </ul> <p>Introducing the <a class="existingWikiWord" href="/nlab/show/nilpotence+theorem">nilpotence theorem</a> in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>:</p> <ul> <li id="DevinatzHopkinsSmith88"> <p><a class="existingWikiWord" href="/nlab/show/Ethan+Devinatz">Ethan Devinatz</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Jeffrey+Smith">Jeffrey Smith</a>, <em>Nilpotence and Stable Homotopy Theory I</em>, Annals of Mathematics Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 207-241 (<a href="https://www.jstor.org/stable/1971440">jstor:1971440</a>)</p> </li> <li id="DevinatzHopkinsSmith98"> <p><a class="existingWikiWord" href="/nlab/show/Ethan+Devinatz">Ethan Devinatz</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Jeffrey+Smith">Jeffrey Smith</a>, <em>Nilpotence and Stable Homotopy Theory II</em>, Annals of Mathematics Second Series, Vol. 148, No. 1 (Jul., 1998), pp. 1-49 (<a href="https://www.jstor.org/stable/120991">jstor:120991</a>)</p> </li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/Conner-Floyd+isomorphism">Conner-Floyd isomorphism</a> for the <a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+orientation">Atiyah-Bott-Shapiro orientation</a> of <a class="existingWikiWord" href="/nlab/show/KU">KU</a> and <a class="existingWikiWord" href="/nlab/show/KO">KO</a> (<a class="existingWikiWord" href="/nlab/show/cobordism+theory+determining+homology+theory">cobordism theory determining homology theory</a>):</p> <ul> <li id="HopkinsHovey92"><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em>Spin cobordism determines real K-theory</em>, Mathematische Zeitschrift volume 210, pages 181–196 (1992) (<a href="https://doi.org/10.1007/BF02571790">doi:10.1007/BF02571790</a>, <a class="existingWikiWord" href="/nlab/files/HopkinsHoveyCobordismK.pdf" title="pdf">pdf</a>)</li> </ul> <p>On generalized (<a class="existingWikiWord" href="/nlab/show/transchromatic+character">transchromatic</a>) <a class="existingWikiWord" href="/nlab/show/group+characters">group characters</a> via <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a>:</p> <ul> <li id="HopkinsKuhnRavenel00"><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Kuhn">Nicholas Kuhn</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <em>Generalized group characters and complex oriented cohomology theories</em>, J. Amer. Math. Soc. <strong>13</strong> (2000) 553-594 [<a href="https://doi.org/10.1090/S0894-0347-00-00332-5">doi:10.1090/S0894-0347-00-00332-5</a>, <a href="http://www.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf">pdf</a>]</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/elliptic+genera">elliptic genera</a>, the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a> and the <a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a>:</p> <ul> <li id="AndoHopkinsStrickland01"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>Elliptic spectra, the Witten genus and the theorem of the cube</em>, Invent. Math. 146 (2001) 595–687 MR1869850 (<a href="https://doi.org/10.1007/s002220100175">doi:10.1007/s002220100175</a>, <a href="http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf">pdf</a>)</p> </li> <li id="AndoHopkinsStrickland02"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>The sigma orientation is an H-infinity map</em>, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 (<a href="http://arxiv.org/abs/math/0204053">arXiv:math/0204053</a>, <a href="https://doi.org/10.1353/ajm.2004.0008">doi:10.1353/ajm.2004.0008</a>)</p> </li> </ul> <p>The construction of <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a> was originally announced, as joint work with <a class="existingWikiWord" href="/nlab/show/Mark+Mahowald">Mark Mahowald</a> and <a class="existingWikiWord" href="/nlab/show/Haynes+Miller">Haynes Miller</a>, in</p> <ul> <li id="Hopkins94"><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, section 9 of <em>Topological modular forms, the Witten Genus, and the theorem of the cube</em>, Proceedings of the International Congress of Mathematics, Zürich 1994 (<a class="existingWikiWord" href="/nlab/files/Hopkins_TopModFormsAtICM.pdf" title="pdf">pdf</a>, <a href="https://doi.org/10.1007/978-3-0348-9078-6_49">doi:10.1007/978-3-0348-9078-6_49</a>)</li> </ul> <p>(There the spectrum was still called “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>eo</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">eo_2</annotation></semantics></math>” instead of “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>tmf</mi></mrow><annotation encoding="application/x-tex">tmf</annotation></semantics></math>”.) The details of the definition then appeared in</p> <ul> <li id="Hopkins02"><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, section 4 of <em>Algebraic topology and modular forms</em>, Proceedings of the ICM, Beijing 2002, vol. 1, 283–309 (<a href="http://arxiv.org/abs/math/0212397">arXiv:math/0212397</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a> and <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a>:</p> <ul> <li id="Hopkins"><a class="existingWikiWord" href="/nlab/show/Mike+Hopkins">Mike Hopkins</a> (talk notes by <a class="existingWikiWord" href="/nlab/show/Michael+Hill">Michael Hill</a>), <em>Stacks and complex oriented cohomology theories</em> (<a href="http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter08/MikesTalk1.pdf">pdf</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/twisted+equivariant+K-theory">twisted equivariant K-theory</a> with an eye towards <a class="existingWikiWord" href="/nlab/show/twisted+ad-equivariant+K-theory">twisted ad-equivariant K-theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>Twisted equivariant K-theory with complex coefficients</em>, Journal of Topology, Volume 1, Issue 1 <p>(<a href="https://arxiv.org/abs/math/0206257">arXiv:math/0206257</a>, <a href="https://doi.org/10.1112/jtopol/jtm001">doi:10.1112/jtopol/jtm001</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/twisted+ad-equivariant+K-theory">twisted ad-equivariant K-theory</a> of <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a> and the identification with the <a class="existingWikiWord" href="/nlab/show/Verlinde+ring">Verlinde ring</a> of <a class="existingWikiWord" href="/nlab/show/positive+energy+representations">positive energy representations</a> of their <a class="existingWikiWord" href="/nlab/show/loop+group">loop group</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+S.+Freed">Daniel S. Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>,</p> <ol> <li> <p><em><a class="existingWikiWord" href="/nlab/show/Loop+Groups+and+Twisted+K-Theory">Loop Groups and Twisted K-Theory</a> I</em>,</p> <p>J. Topology, 4 (2011), 737-789</p> <p>(<a href="http://arxiv.org/abs/0711.1906">arXiv:0711.1906</a>, <a href="https://doi.org/10.1112/jtopol/jtr019">doi:10.1112/jtopol/jtr019</a>)</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/Loop+Groups+and+Twisted+K-Theory">Loop Groups and Twisted K-Theory</a> II</em>,</p> <p>J. Amer. Math. Soc. 26 (2013), 595-644</p> <p>(<a href="http://arxiv.org/abs/math/0511232">arXiv:0511232</a>, <a href="https://doi.org/10.1090/S0894-0347-2013-00761-4">doi:10.1090/S0894-0347-2013-00761-4</a>)</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/Loop+Groups+and+Twisted+K-Theory">Loop Groups and Twisted K-Theory</a> III</em>,</p> <p>Annals of Mathematics, Volume 174 (2011) 947-1007</p> <p>(<a href="http://arxiv.org/abs/math/0312155">arXiv:math/0312155</a>, <a href="http://dx.doi.org/10.4007/annals.2011.174.2.5">doi:10.4007/annals.2011.174.2.5</a>)</p> </li> </ol> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups+of+units">∞-groups of units</a>, <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> and <a class="existingWikiWord" href="/nlab/show/twisted+generalized+cohomology">twisted generalized cohomology</a>:</p> <ul> <li id="ABGHR08"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Units of ring spectra and Thom spectra</em> (<a href="http://arxiv.org/abs/0810.4535">arXiv:0810.4535</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory</em>, Journal of Topology, Volume7, Issue 4, December 2014 (<a href="https://arxiv.org/abs/1403.4320">arXiv:1403.4320</a>, <a href="https://doi.org/10.1112/jtopol/jtu009">arXiv:10.1112/jtopol/jtu009</a>)</p> </li> <li id="ABGHR14"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categorical approach to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-line bundles, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module Thom spectra, and twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-homology</em>, Journal of Topology Volume 7, Issue 3 2014 Pages 869–893 (<a href="http://arxiv.org/abs/1403.4325">arXiv:1403.4325</a>, <a href="https://doi.org/10.1112/jtopol/jtt035">doi:10.1112/jtopol/jtt035</a>)</p> </li> </ul> <p>Solving the <a class="existingWikiWord" href="/nlab/show/Arf-Kervaire+invariant+problem">Arf-Kervaire invariant problem</a> with methods of <a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+theory">equivariant stable homotopy theory</a>:</p> <ul> <li id="HillHopkinsRavenel21"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hill">Michael Hill</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <em>Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem</em>, New Mathematical Monographs, Cambridge University Press (2021) [<a href="https://doi.org/10.1017/9781108917278">doi:10.1017/9781108917278</a>]</p> </li> <li id="HillHopkinsRavenel09"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hill">Michael Hill</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <em>On the non-existence of elements of Kervaire invariant one</em>, Annals of Mathematics <strong>184</strong> 1 (2016)[<a href="https://doi.org/10.4007/annals.2016.184.1.1">doi:10.4007/annals.2016.184.1.1</a>, <a href="http://arxiv.org/abs/0908.3724">arXiv:0908.3724</a>, <a href="https://www.math.rochester.edu/people/faculty/doug/otherpapers/Skye_handout.pdf">talk slides</a>]</p> </li> <li id="HillHopkinsRavenel"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hill">Michael Hill</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <em>The Arf-Kervaire problem in algebraic topology: Sketch of the proof</em>, Current Developments in Mathematics, 2010: 1-44 (2011) (<a class="existingWikiWord" href="/nlab/files/HHRKervaire.pdf" title="pdf">pdf</a>, <a href="https://dx.doi.org/10.4310/CDM.2010.v2010.n1.a1">doi:10.4310/CDM.2010.v2010.n1.a1</a>)</p> </li> <li id="HillHopkinsRavenelIntroduction"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hill">Michael Hill</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <em>The Arf-Kervaire invariant problem in algebraic topology: introduction</em> (2016) [<a href="http://math.ucla.edu/~mikehill/Research/CDMHistory.pdf">pdf</a>]</p> </li> </ul> <p>Introducing <a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a> and its specialization to Hodge-filtered <a class="existingWikiWord" href="/nlab/show/complex+cobordism+theory">complex cobordism theory</a>:</p> <ul> <li id="HopkinsQuick14"><a class="existingWikiWord" href="/nlab/show/Michael+J.+Hopkins">Michael J. Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Gereon+Quick">Gereon Quick</a>, §5 in: <em>Hodge filtered complex bordism</em>, Journal of Topology <strong>8</strong> 1 (2014) 147-183 [<a href="https://arxiv.org/abs/1212.2173">arXiv:1212.2173</a>, <a href="https://doi.org/10.1112/jtopol/jtu021">doi:10.1112/jtopol/jtu021</a>]</li> </ul> <p>On classification of <a class="existingWikiWord" href="/nlab/show/invertible+TQFTs">invertible TQFTs</a> via <a class="existingWikiWord" href="/nlab/show/reflection+positivity">reflection positivity</a>:</p> <ul> <li id="FreedHopkins21"><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <em>Reflection positivity and invertible topological phases</em>, Geometry & Topology <strong>25</strong> (2021) 1165–1330 [<a href="https://arxiv.org/abs/1604.06527">arXiv:1604.06527</a>, <a href="https://doi.org/10.2140/gt.2021.25.1165">doi:10.2140/gt.2021.25.1165</a>]</li> </ul> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A+Survey+of+Elliptic+Cohomology">A Survey of Elliptic Cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopkins-Miller+theorem">Hopkins-Miller theorem</a></p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/people">people</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on June 21, 2024 at 08:32:45. 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