Condensed mathematics is a theory developed by and Peter Scholze which, according to some, aims to unify various mathematical subfields, including

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, complex geometry, and

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

Idea

The fundamental idea in the development of the theory is given by replacing

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

s 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 Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

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

, p-adic analytic geometry and

complex analytic geometry In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) Reduced ring, reduced or complex analytic space i ...

Definition

A ''condensed set'' is a sheaf of sets on the site of

profinite set In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in t ...

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. 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 A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...

. 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 In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated. An example of such a geometry is the Dehn plane. Non-Archimedean geometries may, as the example indicates, have properties ...

into the theory. In 2020 Scholze completed a proof of a result which would enable the incorporation of

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

as well as complex geometry into the condensed mathematics framework, using the notion of ''

liquid vector space The concept of a liquid vector space is part of condensed mathematics. Liquid vector spaces are an alternative to topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviate ...

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 formalized and verified proof. Over a 6-month period, a group led by Johan Commelin verified the central part of the proof using the

proof assistant In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor ...

Lean Lean, leaning or LEAN may refer to: Business practices * Lean thinking, a business methodology adopted in various fields ** Lean construction, an adaption of lean manufacturing principles to the design and construction process ** Lean governmen ...

. As of 14 July 2022, the proof has been completed. Coincidentally, in 2019 Barwick and Haine introduced a very similar theory of ''

pyknotic object In mathematics, especially in topology, a pyknotic set is a sheaf of sets on the site of compact Hausdorff spaces (with some fixed Grothendieck universes). The notion was introduced by Barwick and Haine to provide a convenient setting for homologica ...

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 universes, whereas condensed mathematics can be developed strictly within ZFC.

References

Firther reading

* https://mathoverflow.net/questions/441838/condensed-vs-pyknotic-vs-consequential

External links