condensed mathematics

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 might expect to be able to incorporate algebraic geometry, p-adic analytic geometry and complex analytic geometry.{{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}}

In condensed mathematics, liquid vector spaces are alternatives to topological vector spaces,{{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}}{{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}} the category of which has better abstract properties than that of topological vector spaces. This allows for more abstract approaches using tools such as abelian categories.

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 $\backslash underline\; X$, which to any profinite set $S$ associates the set of continuous maps $S\backslash to\; X$. If $X$ is a topological group or ring, then $\backslash underline\; X$ is a condensed group or ring.

In 2013, Bhargav Bhatt and Peter Scholze introduced a general notion of pro-étale site associated to an arbitrary scheme. In 2018, Dustin Clausen and Scholze arrived at the conclusion that the pro-étale site of a single point, which is isomorphic to the site of profinite sets introduced above, already 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.{{cite web|url=https://www.math.uni-bonn.de/people/scholze/Condensed.pdf |title=Lectures on Condensed Mathematics|last=Scholze|first=Peter|year= 2019}}

In 2020 Scholze completed a proof of their results 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.{{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}} Over a 6-month period, a group led by Johan Commelin verified the central part of the proof using the proof assistant Lean.{{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}}{{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 }} As of 14 July 2022, the proof has been completed.{{Cite web|title=leanprover-community/lean-liquid|url=https://github.com/leanprover-community/lean-liquid/|access-date=2022-07-14|website=Github|language=en}}

Coincidentally, in 2019 Barwick and Haine introduced a 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.{{cite web|url=https://ncatlab.org/nlab/show/pyknotic+set|title=Pyknotic sets|website=nLab}}

• Liquid vector space
• Pyknotic set

• https://mathoverflow.net/questions/441838/condensed-vs-pyknotic-vs-consequential
• https://mathoverflow.net/questions/tagged/condensed-mathematics
• https://math.stackexchange.com/questions/4044728/examples-of-the-difference-between-topological-spaces-and-condensed-sets

• {{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=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}}
• {{cite web |title=Notes on condensed mathematics |url=https://kskedlaya.org/condensed/condensed.html |website=Notes on condensed mathematics - Kiran S. Kedlaya}}

Category:Topology

Category:Algebraic geometry

Category:Analytic geometry

Category:Functional analysis