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href="/wiki/Peter_Scholze" title="Peter Scholze">Peter Scholze</a> which, according to some,<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag possibly uses too-vague attribution or weasel words. (July 2024)">who?</span></a></i>]</sup> aims to unify various <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> subfields, including <a href="/wiki/Topology" title="Topology">topology</a>, <a href="/wiki/Complex_geometry" title="Complex geometry">complex geometry</a>, and <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2024)">citation needed</span></a></i>]</sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Idea">Idea</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condensed_mathematics&action=edit&section=1" title="Edit section: Idea"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fundamental idea in the development of the theory is given by replacing <a href="/wiki/Topological_space" title="Topological space">topological spaces</a> by <i>condensed sets</i>, defined below. The <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of condensed sets, as well as related categories such as that of condensed <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a>, are much better behaved than the <a href="/wiki/Category_of_topological_spaces" title="Category of topological spaces">category of topological spaces</a>. In particular, unlike the category of <a href="/wiki/Topological_group" title="Topological group">topological abelian groups</a>, the category of condensed abelian groups is an <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a>, which allows for the use of tools from <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a> in the study of those structures. </p><p>The framework of condensed mathematics turns out to be general enough that, by considering various "spaces" with <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves</a> valued in condensed algebras, one is able to incorporate <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, <a href="/wiki/Rigid_analytic_space" title="Rigid analytic space"><i>p</i>-adic analytic geometry</a> and <a href="/wiki/Complex_analytic_variety" title="Complex analytic variety">complex analytic geometry</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Liquid_vector_space">Liquid vector space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condensed_mathematics&action=edit&section=2" title="Edit section: Liquid vector space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In condensed mathematics, <b>liquid vector spaces</b> are alternatives to <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condensed_mathematics&action=edit&section=3" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>condensed set</i> is a <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> of sets on the <a href="/wiki/Site_(mathematics)" class="mw-redirect" title="Site (mathematics)">site</a> of <a href="/wiki/Profinite_set" class="mw-redirect" title="Profinite set">profinite sets</a>, with the <a href="/wiki/Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a> given by finite, jointly surjective collections of maps. Similarly, a <i>condensed group</i>, <i>condensed ring</i>, etc. is defined as a sheaf of groups, rings etc. on this site. </p><p>To any topological space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> one can associate a condensed set, customarily denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>X</mi> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/795941c3bd441d39b514ee92a014d6dd031dde66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.036ex; margin-bottom: -0.827ex; width:2.017ex; height:3.176ex;" alt="{\displaystyle {\underline {X}}}"></span>, which to any profinite set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> associates the set of continuous maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5566629251250ba644683a256f3ae6b6ec516d8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.093ex; height:2.176ex;" alt="{\displaystyle S\to X}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is a topological group or ring, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>X</mi> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/795941c3bd441d39b514ee92a014d6dd031dde66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.036ex; margin-bottom: -0.827ex; width:2.017ex; height:3.176ex;" alt="{\displaystyle {\underline {X}}}"></span> is a condensed group or ring. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condensed_mathematics&action=edit&section=4" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2013, <a href="/wiki/Bhargav_Bhatt_(mathematician)" title="Bhargav Bhatt (mathematician)">Bhargav Bhatt</a> and <a href="/wiki/Peter_Scholze" title="Peter Scholze">Peter Scholze</a> introduced a general notion of <i>pro-<a href="/wiki/%C3%89tale_topology" title="Étale topology">étale</a> site</i> associated to an arbitrary <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a>. In 2018, together with Dustin Clausen, they arrived at the conclusion that already the pro-étale site of a single point, which is isomorphic to the site of profinite sets introduced above, has rich enough structure to <a href="/wiki/Embedding" title="Embedding">realize</a> large classes of topological spaces as sheaves on it. Further developments have led to a theory of condensed sets and <i>solid abelian groups</i>, through which one is able to incorporate <a href="/wiki/Non-Archimedean_geometry" title="Non-Archimedean geometry">non-Archimedean geometry</a> into the theory.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In 2020 Scholze completed a proof of a result which would enable the incorporation of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> as well as complex geometry into the condensed mathematics framework, using the notion of <i><a href="/wiki/Liquid_vector_space" class="mw-redirect" title="Liquid vector space">liquid vector spaces</a></i>. The argument has turned out to be quite subtle, and to get rid of any doubts about the validity of the result, he asked other mathematicians to provide a <a href="/wiki/Formal_proof" title="Formal proof">formalized and verified proof</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Quanta21_6-0" class="reference"><a href="#cite_note-Quanta21-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Over a 6-month period, a group led by Johan Commelin verified the central part of the proof using the <a href="/wiki/Proof_assistant" title="Proof assistant">proof assistant</a> <a href="/wiki/Lean_(proof_assistant)" title="Lean (proof assistant)">Lean</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Quanta21_6-1" class="reference"><a href="#cite_note-Quanta21-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> As of 14 July 2022, the proof has been completed.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Coincidentally, in 2019 Barwick and Haine introduced a very similar theory of <i><a href="/wiki/Pyknotic_object" class="mw-redirect" title="Pyknotic object">pyknotic objects</a></i>. This theory is very closely related to that of condensed sets, with the main differences being set-theoretic in nature: pyknotic theory depends on a choice of <a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck universes</a>, whereas condensed mathematics can be developed strictly within <a href="/wiki/ZFC" class="mw-redirect" title="ZFC">ZFC</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condensed_mathematics&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Liquid_vector_space" class="mw-redirect" title="Liquid vector space">Liquid vector space</a></li> <li><a href="/wiki/Pyknotic_set" title="Pyknotic set">Pyknotic set</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condensed_mathematics&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFClausenScholze2022" class="citation web cs1">Clausen, Dustin; Scholze, Peter (2022). <a rel="nofollow" class="external text" href="https://people.mpim-bonn.mpg.de/scholze/Complex.pdf">"Condensed Mathematics and Complex Geometry"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Condensed+Mathematics+and+Complex+Geometry&rft.date=2022&rft.aulast=Clausen&rft.aufirst=Dustin&rft.au=Scholze%2C+Peter&rft_id=https%3A%2F%2Fpeople.mpim-bonn.mpg.de%2Fscholze%2FComplex.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/liquid+vector+space">"liquid vector space in nLab"</a>. <i>ncatlab.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-11-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ncatlab.org&rft.atitle=liquid+vector+space+in+nLab&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fliquid%2Bvector%2Bspace&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScholze" class="citation web cs1">Scholze, Peter. <a rel="nofollow" class="external text" href="https://www.math.uni-bonn.de/people/scholze/Analytic.pdf">"Lectures on Analytic Geometry: Lecture III: Condensed ℝ-vector spaces"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">7 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lectures+on+Analytic+Geometry%3A+Lecture+III%3A+Condensed+%E2%84%9D-vector+spaces&rft.aulast=Scholze&rft.aufirst=Peter&rft_id=https%3A%2F%2Fwww.math.uni-bonn.de%2Fpeople%2Fscholze%2FAnalytic.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScholze2019" class="citation web cs1">Scholze, Peter (2019). <a rel="nofollow" class="external text" href="https://www.math.uni-bonn.de/people/scholze/Condensed.pdf">"Lectures on Condensed Mathematics"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lectures+on+Condensed+Mathematics&rft.date=2019&rft.aulast=Scholze&rft.aufirst=Peter&rft_id=https%3A%2F%2Fwww.math.uni-bonn.de%2Fpeople%2Fscholze%2FCondensed.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScholze2020" class="citation web cs1">Scholze, Peter (2020-12-05). <a rel="nofollow" class="external text" href="https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-experiment/">"Liquid tensor experiment"</a>. <i>Xena</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-06-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Xena&rft.atitle=Liquid+tensor+experiment&rft.date=2020-12-05&rft.aulast=Scholze&rft.aufirst=Peter&rft_id=https%3A%2F%2Fxenaproject.wordpress.com%2F2020%2F12%2F05%2Fliquid-tensor-experiment%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-Quanta21-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Quanta21_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Quanta21_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartnett2021" class="citation news cs1">Hartnett, Kevin (July 28, 2021). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/lean-computer-program-confirms-peter-scholze-proof-20210728/">"Proof Assistant Makes Jump to Big-League Math"</a>. <i><a href="/wiki/Quanta_Magazine" title="Quanta Magazine">Quanta Magazine</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quanta+Magazine&rft.atitle=Proof+Assistant+Makes+Jump+to+Big-League+Math&rft.date=2021-07-28&rft.aulast=Hartnett&rft.aufirst=Kevin&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Flean-computer-program-confirms-peter-scholze-proof-20210728%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScholze2021" class="citation web cs1">Scholze, Peter (2021-06-05). <a rel="nofollow" class="external text" href="https://xenaproject.wordpress.com/2021/06/05/half-a-year-of-the-liquid-tensor-experiment-amazing-developments//">"Half a year of the Liquid Tensor Experiment: Amazing developments"</a>. <i>Xena</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-06-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Xena&rft.atitle=Half+a+year+of+the+Liquid+Tensor+Experiment%3A+Amazing+developments&rft.date=2021-06-05&rft.aulast=Scholze&rft.aufirst=Peter&rft_id=https%3A%2F%2Fxenaproject.wordpress.com%2F2021%2F06%2F05%2Fhalf-a-year-of-the-liquid-tensor-experiment-amazing-developments%2F%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://github.com/leanprover-community/lean-liquid/">"leanprover-community/lean-liquid"</a>. <i>Github</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-07-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Github&rft.atitle=leanprover-community%2Flean-liquid&rft_id=https%3A%2F%2Fgithub.com%2Fleanprover-community%2Flean-liquid%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/pyknotic+set">"Pyknotic sets"</a>. <i>nLab</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Pyknotic+sets&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fpyknotic%2Bset&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condensed_mathematics&action=edit&section=7" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external free" href="https://mathoverflow.net/questions/441838/condensed-vs-pyknotic-vs-consequential">https://mathoverflow.net/questions/441838/condensed-vs-pyknotic-vs-consequential</a></li> <li><a rel="nofollow" class="external free" href="https://mathoverflow.net/questions/tagged/condensed-mathematics">https://mathoverflow.net/questions/tagged/condensed-mathematics</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condensed_mathematics&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScholze2019" class="citation web cs1">Scholze, Peter (2019). <a rel="nofollow" class="external text" href="https://www.math.uni-bonn.de/people/scholze/Condensed.pdf">"Lectures on Condensed Mathematics"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lectures+on+Condensed+Mathematics&rft.date=2019&rft.aulast=Scholze&rft.aufirst=Peter&rft_id=https%3A%2F%2Fwww.math.uni-bonn.de%2Fpeople%2Fscholze%2FCondensed.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScholze2020" class="citation web cs1">Scholze, Peter (2020). <a rel="nofollow" class="external text" href="https://www.math.uni-bonn.de/people/scholze/Analytic.pdf">"Lectures on Analytic Geometry"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lectures+on+Analytic+Geometry&rft.date=2020&rft.aulast=Scholze&rft.aufirst=Peter&rft_id=https%3A%2F%2Fwww.math.uni-bonn.de%2Fpeople%2Fscholze%2FAnalytic.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClausenScholze2022" class="citation web cs1">Clausen, Dustin; Scholze, Peter (2022). <a rel="nofollow" class="external text" href="https://people.mpim-bonn.mpg.de/scholze/Complex.pdf">"Condensed Mathematics and Complex Geometry"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Condensed+Mathematics+and+Complex+Geometry&rft.date=2022&rft.aulast=Clausen&rft.aufirst=Dustin&rft.au=Scholze%2C+Peter&rft_id=https%3A%2F%2Fpeople.mpim-bonn.mpg.de%2Fscholze%2FComplex.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPstrągowski2020" class="citation web cs1">Pstrągowski, Piotr Tadeusz (2020-11-09). <a rel="nofollow" class="external text" href="https://www.math.ku.dk/english/calendar/events/condensed-mathematics/">"Masterclass in Condensed Mathematics"</a>. <i>www.math.ku.dk</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-06-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.math.ku.dk&rft.atitle=Masterclass+in+Condensed+Mathematics&rft.date=2020-11-09&rft.aulast=Pstr%C4%85gowski&rft.aufirst=Piotr+Tadeusz&rft_id=https%3A%2F%2Fwww.math.ku.dk%2Fenglish%2Fcalendar%2Fevents%2Fcondensed-mathematics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.uchicago.edu/~may/REU2023/Condensed.pdf">"Notes on condensed mathematics"</a> <span class="cs1-format">(PDF)</span>. <i>The University of Chicago Mathematics REU 2023</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+University+of+Chicago+Mathematics+REU+2023&rft.atitle=Notes+on+condensed+mathematics&rft_id=https%3A%2F%2Fmath.uchicago.edu%2F~may%2FREU2023%2FCondensed.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondensed+mathematics" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Condensed_mathematics&oldid=1253717874">https://en.wikipedia.org/w/index.php?title=Condensed_mathematics&oldid=1253717874</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Topology" title="Category:Topology">Topology</a></li><li><a href="/wiki/Category:Algebraic_geometry" title="Category:Algebraic geometry">Algebraic geometry</a></li><li><a href="/wiki/Category:Analytic_geometry" title="Category:Analytic geometry">Analytic geometry</a></li><li><a href="/wiki/Category:Functional_analysis" title="Category:Functional analysis">Functional analysis</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_specifically_marked_weasel-worded_phrases" title="Category:All articles with specifically marked weasel-worded phrases">All articles with specifically marked weasel-worded phrases</a></li><li><a href="/wiki/Category:Articles_with_specifically_marked_weasel-worded_phrases_from_July_2024" title="Category:Articles with specifically marked weasel-worded phrases from July 2024">Articles with specifically marked weasel-worded phrases from July 2024</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_July_2024" title="Category:Articles with unsourced statements from July 2024">Articles with unsourced statements from July 2024</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 October 2024, at 14:25<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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