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Condensed mathematics is a theory developed by Dustin Clausen and Peter Scholze which, according to some,^[who?] aims to unify various mathematical subfields, including topology, complex geometry, and algebraic geometry.^{[citation needed]}

Idea

The fundamental idea in the development of the theory is given by replacing topological spaces by condensed sets, defined below. The category of condensed sets, as well as related categories such as that of condensed abelian groups, are much better behaved than the category of topological spaces. In particular, unlike the category of topological abelian groups, the category of condensed abelian groups is an abelian category, which allows for the use of tools from homological algebra in the study of those structures.

The framework of condensed mathematics turns out to be general enough that, by considering various "spaces" with sheaves valued in condensed algebras, one is able to incorporate algebraic geometry, p-adic analytic geometry and complex analytic geometry.^[1]

Liquid vector space

In condensed mathematics, liquid vector spaces are alternatives to topological vector spaces.^[2]^[3]

Definition

A condensed set is a sheaf of sets on the site of profinite sets, with the Grothendieck topology given by finite, jointly surjective collections of maps. Similarly, a condensed group, condensed ring, etc. is defined as a sheaf of groups, rings etc. on this site.

To any topological space $X$ one can associate a condensed set, customarily denoted ${\underline {X}}$ , which to any profinite set $S$ associates the set of continuous maps $S\to X$ . If $X$ is a topological group or ring, then ${\underline {X}}$ is a condensed group or ring.

History

In 2013, Bhargav Bhatt and Peter Scholze introduced a general notion of pro-étale site associated to an arbitrary scheme. 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 realize large classes of topological spaces as sheaves on it. Further developments have led to a theory of condensed sets and solid abelian groups, through which one is able to incorporate non-Archimedean geometry into the theory.^[4]

In 2020 Scholze completed a proof of a result which would enable the incorporation of functional analysis as well as complex geometry into the condensed mathematics framework, using the notion of liquid vector spaces. 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 formalized and verified proof.^[5]^[6] Over a 6-month period, a group led by Johan Commelin verified the central part of the proof using the proof assistant Lean.^[7]^[6] As of 14 July 2022, the proof has been completed.^[8]

Coincidentally, in 2019 Barwick and Haine introduced a very similar theory of pyknotic objects. 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 Grothendieck universes, whereas condensed mathematics can be developed strictly within ZFC.^[9]

References

^ Clausen, Dustin; Scholze, Peter (2022). "Condensed Mathematics and Complex Geometry" (PDF).
^ "liquid vector space in nLab". ncatlab.org. Retrieved 2023-11-07.
^ Scholze, Peter. "Lectures on Analytic Geometry: Lecture III: Condensed ℝ-vector spaces" (PDF). Retrieved 7 November 2023.
^ Scholze, Peter (2019). "Lectures on Condensed Mathematics" (PDF).
^ Scholze, Peter (2020-12-05). "Liquid tensor experiment". Xena. Retrieved 2022-06-28.
^ ^a ^b Hartnett, Kevin (July 28, 2021). "Proof Assistant Makes Jump to Big-League Math". Quanta Magazine.
^ Scholze, Peter (2021-06-05). "Half a year of the Liquid Tensor Experiment: Amazing developments". Xena. Retrieved 2022-06-28.
^ "leanprover-community/lean-liquid". Github. Retrieved 2022-07-14.
^ "Pyknotic sets". nLab.

External links

Scholze, Peter (2019). "Lectures on Condensed Mathematics" (PDF).
Scholze, Peter (2020). "Lectures on Analytic Geometry" (PDF).
Clausen, Dustin; Scholze, Peter (2022). "Condensed Mathematics and Complex Geometry" (PDF).
Pstrągowski, Piotr Tadeusz (2020-11-09). "Masterclass in Condensed Mathematics". www.math.ku.dk. Retrieved 2021-06-21.
"Notes on condensed mathematics" (PDF). The University of Chicago Mathematics REU 2023.

[1] Clausen, Dustin; Scholze, Peter (2022). "Condensed Mathematics and Complex Geometry" (PDF).

[2] "liquid vector space in nLab". ncatlab.org. Retrieved 2023-11-07.

[3] Scholze, Peter. "Lectures on Analytic Geometry: Lecture III: Condensed ℝ-vector spaces" (PDF). Retrieved 7 November 2023.

[4] Scholze, Peter (2019). "Lectures on Condensed Mathematics" (PDF).

[5] Scholze, Peter (2020-12-05). "Liquid tensor experiment". Xena. Retrieved 2022-06-28.

[Quanta21-6] Hartnett, Kevin (July 28, 2021). "Proof Assistant Makes Jump to Big-League Math". Quanta Magazine.

[7] Scholze, Peter (2021-06-05). "Half a year of the Liquid Tensor Experiment: Amazing developments". Xena. Retrieved 2022-06-28.

[8] "leanprover-community/lean-liquid". Github. Retrieved 2022-07-14.

[9] "Pyknotic sets". nLab.

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+{{Short description|Area of mathematics using condensed sets}}
-'''Condensed mathematics''' is the (potential) unification of various [[mathematics|mathematical]] subfields, including [[topology]], [[geometry]], and [[number theory]]. It  asserts that analogs in the individual fields are instead different expressions of the same concepts (similarly to the way in which different human languages can express the same thing).<ref name=":0">{{Cite journal|last=Castelvecchi|first=Davide|date=2021-06-18|title=Mathematicians welcome computer-assisted proof in ‘grand unification’ theory|url=https://www.nature.com/articles/d41586-021-01627-2|journal=Nature|language=en|doi=10.1038/d41586-021-01627-2}}</ref>
+'''Condensed mathematics''' is a theory developed by [[Dustin Clausen]] and [[Peter Scholze]] which, according to some,{{who|date=July 2024}} aims to unify various [[mathematics|mathematical]] subfields, including [[topology]], [[complex geometry]], and [[algebraic geometry]].{{fact|date=July 2024}}
-== Concept ==
+== Idea ==
-Topology plays a crucial part in number theory, [[functional analysis]], and geometry. Topologically, a coffee cup with a handle has the same topology as a doughnut, while one without has the same topology as a [[sphere]].
+The fundamental idea in the development of the theory is given by replacing [[topological space]]s by ''condensed sets'', defined below. The [[category (mathematics)|category]] of condensed sets, as well as related categories such as that of condensed [[abelian group]]s, are much better behaved than the [[category of topological spaces]]. In particular, unlike the category of [[topological group|topological abelian groups]], the category of condensed abelian groups is an [[abelian category]], which allows for the use of tools from [[homological algebra]] in the study of those structures.
-Many mathematical objects have a topology — a way of measuring the proximity of the object's parts. These include [[topological abelian group]]s and [[topological vector space]]s. [[function (mathematics)|Functions]] typically can be represented in spaces with an infinite number of dimensions. The set of [[real number]]s has the topology of a straight line, while [[p-adic number|''p''-adic number]] systems have a [[fractal]] topology.<ref name=":0" /><ref>{{Cite journal|last=Ferrari|first=S.|date=December 2019|title=Sobolev spaces with respect to a weighted Gaussian measure in infinite dimensions|url=http://dx.doi.org/10.1142/s0219025719500267|journal=Infinite Dimensional Analysis, Quantum Probability and Related Topics|volume=22|issue=04|pages=1950026|doi=10.1142/s0219025719500267|issn=0219-0257|arxiv=1510.08283}}</ref>
+The framework of condensed mathematics turns out to be general enough that, by considering various "spaces" with [[sheaf (mathematics)|sheaves]] valued in condensed algebras, one is able to incorporate [[algebraic geometry]], [[rigid analytic space|''p''-adic analytic geometry]] and [[complex analytic variety|complex analytic geometry]].<ref>{{cite web|url=https://people.mpim-bonn.mpg.de/scholze/Complex.pdf|title=Condensed Mathematics and Complex Geometry|last1=Clausen|first1=Dustin|last2=Scholze|first2=Peter|year= 2022}}</ref>
-=== Subgoals ===
-Parts of the unification program are to replace [[topological space]]s by “condensed sets",<ref name=":1">{{Cite web|title=Condensed mathematics – SPP 2026|url=https://blog.spp2026.de/condensed-mathematics/|access-date=2021-06-21|first=Steffen |last=Kionke |date=17 December 2020|language=en-GB}}</ref><ref name=":2">{{Cite web|last=Scholze|first=Peter|date=2020-12-05|title=Liquid tensor experiment|url=https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-experiment/|url-status=live|website=Xena|language=en}}</ref> turn functional analysis into a branch of [[commutative algebra]], and various types of [[analytic geometry]] into [[algebraic geometry]].<ref>{{Cite web|title=condensed mathematics in nLab|url=https://ncatlab.org/nlab/show/condensed+mathematics|access-date=2021-06-21|website=ncatlab.org}}</ref> Condensed sets form condensed [[abelian group]]s into an [[abelian category]], which satisfies the constraints of [[homological algebra]].<ref name=":1" />
+===Liquid vector space===
-Approximately, a topological space ''X'' is replaced by the [[functor]] that takes a [[profinite set]] ''S'' to the set of continuous maps from ''S'' to ''X''. For every [[compact space|compact]] [[Hausdorff space]] ''X'' there is a [[surjection]] from a profinite set onto ''X''. For example, one such surjection is the decimal expansion from the profinite set of [[sequence]]s in {0,1,2,3,4,5,6,7,8,9} onto the [[interval (mathematics)|interval]] [0,&thinsp;1].<ref name=":1" />
+In condensed mathematics, '''liquid vector spaces''' are alternatives to [[topological vector space]]s.<ref>{{Cite web |title=liquid vector space in nLab |url=https://ncatlab.org/nlab/show/liquid+vector+space |access-date=2023-11-07 |website=ncatlab.org}}</ref><ref>{{Cite web |last=Scholze |first=Peter |title=Lectures on Analytic Geometry: Lecture III: Condensed ℝ-vector spaces |url=https://www.math.uni-bonn.de/people/scholze/Analytic.pdf |access-date=7 November 2023}}</ref>
+==Definition==
-== History ==<ref>University of Copenhagen </ref>
-In 2018, German [[number theory|number theorist]] and [[Fields Medal]] winner [[Peter Scholze]] and Canadian/American mathematician Dustin Clausen realized that conventional topology exposed incompatibilities across geometry, functional analysis and ''p''-adic numbers — and became convinced that alternative foundations could bridge those gaps. The pair announced their effort to rebuild mathematics in 2019.<ref name=":0" />
+A ''condensed set'' is a [[Sheaf (mathematics)|sheaf]] of sets on the [[site (mathematics)|site]] of [[profinite set]]s, with the [[Grothendieck topology]] given by finite, jointly surjective collections of maps. Similarly, a ''condensed group'', ''condensed ring'', etc. is defined as a sheaf of groups, rings etc. on this site.
-In 2020 Scholze authored an involved [[mathematical proof|proof]] that was critical to the theory. He asked other mathematicians led by Johan Commelin to provide a [[Formal proof|formalized and verified proof]] of this result.<ref name=":2" /> Over a 6-month period the group verified the proof using the [[proof assistant]] [[Lean (proof assistant)
-|Lean]].<ref name=":0" />
+To any topological space <math>X</math> one can associate a condensed set, customarily denoted <math>\underline X</math>, which to any profinite set <math>S</math> associates the set of continuous maps <math>S\to X</math>. If <math>X</math> is a topological group or ring, then  <math>\underline X</math> is a condensed group or ring.
+== History ==
+In 2013, [[Bhargav Bhatt (mathematician)|Bhargav Bhatt]] and [[Peter Scholze]] introduced a general notion of ''pro-[[étale topology|étale]] site'' associated to an arbitrary [[scheme (mathematics)|scheme]]. 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 [[embedding|realize]] large classes of topological spaces as sheaves on it. Further developments have led to a theory of condensed sets and ''solid abelian groups'', through which one is able to incorporate [[non-Archimedean geometry]] into the theory.<ref>{{cite web|url=https://www.math.uni-bonn.de/people/scholze/Condensed.pdf |title=Lectures on Condensed Mathematics|last=Scholze|first=Peter|year= 2019}}</ref>
+In 2020 Scholze completed a proof of a result which would enable the incorporation of [[functional analysis]] as well as complex geometry into the condensed mathematics framework, using the notion of ''[[liquid vector space]]s''. 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 [[Formal proof|formalized and verified proof]].<ref>{{Cite web|last=Scholze|first=Peter|date=2020-12-05|title=Liquid tensor experiment|url=https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-experiment/|access-date=2022-06-28|website=Xena|language=en}}</ref><ref name="Quanta21"/> Over a 6-month period, a group led by Johan Commelin verified the central part of the proof using the [[proof assistant]] [[Lean (proof assistant)|Lean]].<ref>{{Cite web|last=Scholze|first=Peter|date=2021-06-05|title=Half a year of the Liquid Tensor Experiment: Amazing developments|url=https://xenaproject.wordpress.com/2021/06/05/half-a-year-of-the-liquid-tensor-experiment-amazing-developments//|access-date=2022-06-28|website=Xena|language=en}}</ref><ref name="Quanta21">{{cite news |url=https://www.quantamagazine.org/lean-computer-program-confirms-peter-scholze-proof-20210728/ |title=Proof Assistant Makes Jump to Big-League Math |date=July 28, 2021 |first=Kevin |last=Hartnett |work=[[Quanta Magazine]] }}</ref> As of 14 July 2022, the proof has been completed.<ref>{{Cite web|title=leanprover-community/lean-liquid|url=https://github.com/leanprover-community/lean-liquid/|access-date=2022-07-14|website=Github|language=en}}</ref>
+Coincidentally, in 2019 Barwick and Haine introduced a very similar theory of ''[[pyknotic object]]s''. 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 [[Grothendieck universe]]s, whereas condensed mathematics can be developed strictly within [[ZFC]].<ref>{{cite web|url=https://ncatlab.org/nlab/show/pyknotic+set|title=Pyknotic sets|website=nLab}}</ref>
+== See also ==
+* [[Liquid vector space]]
+* [[Pyknotic set]]
 == References ==
 <references/>
+== Further reading ==
+* https://mathoverflow.net/questions/441838/condensed-vs-pyknotic-vs-consequential
+* https://mathoverflow.net/questions/tagged/condensed-mathematics
 == External links ==
-* {{Cite web|last=Scholze|first=Peter|date=2020-12-05|title=Liquid tensor experiment|url=https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-experiment/|access-date=2021-06-21|website=Xena|language=en}}
 * {{cite web|url=https://www.math.uni-bonn.de/people/scholze/Condensed.pdf |title=Lectures on Condensed Mathematics|last=Scholze|first=Peter|year= 2019}}
 * {{cite web|url=https://www.math.uni-bonn.de/people/scholze/Analytic.pdf |title=Lectures on Analytic Geometry|last=Scholze|first=Peter|year= 2020}}
+* {{cite web|url=https://people.mpim-bonn.mpg.de/scholze/Complex.pdf|title=Condensed Mathematics and Complex Geometry|last1=Clausen|first1=Dustin|last2=Scholze|first2=Peter|year= 2022}}
-* {{Cite web|last=Pstragowski|first=Piotr Tadeusz|date=2020-11-09|title=Masterclass in Condensed Mathematics|url=https://www.math.ku.dk/english/calendar/events/condensed-mathematics/|access-date=2021-06-21|website=www.math.ku.dk|language=en}}
+* {{Cite web|last=Pstrągowski|first=Piotr Tadeusz|date=2020-11-09|title=Masterclass in Condensed Mathematics|url=https://www.math.ku.dk/english/calendar/events/condensed-mathematics/|access-date=2021-06-21|website=www.math.ku.dk|language=en}}
+*{{cite web |title=Notes on condensed mathematics |url=https://math.uchicago.edu/~may/REU2023/Condensed.pdf |website=The University of Chicago Mathematics REU 2023}}
 [[Category:Topology]]
+[[Category:Algebraic geometry]]
+[[Category:Analytic geometry]]
+[[Category:Functional analysis]]