Analytic Geometry Scholze
Hermalindo Lepicier
Recently, Peter Scholze posted a challenge to the Xena project to formalize a result in condensed mathematics. The motivation for this result is to provide a new foundation for analysis, not in terms of topology but using a different approach based off of algebraic geometry.
For some reason, this secret plot has so far stopped short of taking over analysis. The goal of this course is to launch a new attack, turning functional analysis into a branch of commutative algebra, and various types of analytic geometry (like manifolds) into algebraic geometry.
As I hope was clear from the phrasing, the opening paragraphs from my
course were not at all serious (I mean, Mumford was joking, so was I). I
have no ambitions to overtake analysis, I just want to be able to talk
about those parts of analysis that I care most about in a way that
combines seamlessly with how arguments work in arithmetic/algebraic
geometry. The hope that this could be done resulted from having had to
do quite a bit of p-adic analysis, and this never felt much like
analysis to me.
What would be nice to know is if this new formalism sheds any similar light on the construction of the liquid tensor product(s), making the theory and the proofs easier. I've watched I think nearly all the lecture series mentioned above, but I don't recall this arising or being mentioned. What was discussed was a new completeness condition analogous to solid/liquid, called "gaseous", but I don't know if this helps with the liquid theory at all as far as simplifying what the Liquid Tensor Experiment needed to show.
Unlike the previous lecture courses on condensed mathematics, there is no running set of lecture notes, but rather Clausen and Scholze are apparently writing a book on this material. So there's no extra written material to check over.
For many (but definitely not all) applications to geometry over the real numbers, the gaseous real vector spaces work just as well, and their theory is much easier to get off the ground than liquid real vector spaces. (Roughly speaking, complex- or real-analytic spaces are fine with gaseous vector spaces, smooth manifolds not so much. The reason is that tensor products of spaces of holomorphic or real-analytic functions behave correctly under the gaseous tensor product, but tensor products of spaces of $C^\infty$-functions are only correct under the liquid tensor product.) This is the route we've taken in the course.
We believe that there is a way to characterize (light) liquid real vector spaces in a way close to how we characterize solid or gaseous modules, in terms of certain endomorphisms of this free module $P$ on a nullsequence to become isomorphisms. However, in this case we are not able to see any simple way to compute the resulting completion functor.
A little fixation of mine has been this one: among some of the most important kinds of geometries there's the theory of real differentiable manifold, complex analytic geometry, schemes, and even more, like different kinds of nonarchimedean objects which I'm quite ignorant about, but are quite similar in spirit to the first ones in my understanding.
Having said that, it seemed however that providing a single formalism to describe them all together is an impossible task (to me at least). I was rather surprised that Scholze's machinery of condensed set, groups, etc, tries indeed to run in this direction.However, at a first glance at the introduction of the subject (more precisely, I'm referring to Clausen and Scholze's notes Condensed mathematics and complex geometry) he points out to "subtle behavior of the real numbers" preventing the notion of analytic ring to give the expected result in the real case.
So now, to those more knowleageable than me on the argument, is this an obstacle getting in the way of us trying to include the theory of real differentiable manifolds in this setting, or are there instead ways to do it with condensed math?Since this is a major point of intrrest in condensed math for me I'd like to understand it better before diving in more deeply in the "cathegorical nonsense".
The purpose of this course is to propose new foundations for analytic geometry. The topics covered are as follows:
1. Light condensed abelian groups.
2. Analytic rings.
3. Analytic stacks.
4. Examples.
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]
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]
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.
In 2013, Bhargav Bhatt and Peter Scholze introduced a general notion of pro-tale site associated to an arbitrary scheme. In 2018, together with Dustin Clausen, they arrived at the conclusion that already the pro-tale 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.[2]
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.[3][4] Over a 6-month period, a group led by Johan Commelin verified the central part of the proof using the proof assistant Lean.[5][4] As of 14 July 2022, the proof has been completed.[6]
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.[7]
Fall quarter 1 Sept. 26 Conrad Motivation for adic spaces .pdf 2 Oct. 3 Conrad Spaces of valuations .pdf 3 Oct. 10 Conrad Spectral spaces and constructible sets .pdf 4 Oct. 17 Conrad Specialization constructions .pdf 5 Oct. 24 Conrad Huber rings and continuous valuations .pdf 6 Oct. 31 Conrad More constructions with Huber rings .pdf 7 Nov. 7 Conrad Rational domains, Tate rings, and analytic points. .pdf 8 Nov. 14 Conrad More properties for Cont(A) .pdf 9 Nov. 21 Conrad Spectrality of Cont(A) .pdf 10 Dec. 5 Conrad Affinoid adic spaces I .pdf 11 Dec. 12 Conrad Affinoid adic spaces II .pdf Winter quarter 12 Jan. 5 Conrad Structure presheaves and rational domains .pdf 13 Jan. 12 Evan Uniformity and sheaf properties .pdf 14 Jan. 19 Conrad Basic generalities on adic spaces .pdf 15 Jan. 26 Conrad Points and lft morphisms .pdf 16 Feb. 2 Conrad Rigid geometry and perfectoid rings .pdf 17 Feb. 9 Conrad The tilting functor .pdf Spring quarter 18 April 8-May 11 Masullo Foundations of perfectoid geometry, I (being edited) 19 May 18-June 1 Masullo Foundations of perfectoid geometry, II
During the 2023-24 academic year the School will have a special program on the $p$-adic arithmetic geometry, organized by Professors Bhargav Bhatt and Jacob Lurie.
Confirmed participants include: Pierre Colmez, Johan DeJong, Ofer Gabber, Lars Hesselholt, Kiran Kedlaya, Matthew Morrow, Wieslawa Niziol, Peter Scholze, Annette Werner and Xinwen Zhu.
The last decade has witnessed some remarkable foundational advances in $p$-adic arithmetic geometry (e.g., the creation of perfectoid geometry and the ensuing reorganization of $p$-adic Hodge theory). These advances have already led to breakthroughs in multiple different areas of mathematics (e.g., significant progress in the Langlands program and the resolution of multiple long-standing conjectures in commutative algebra), have uncovered new phenomena that merit further investigation (e.g., the discovery of new structures on algebraic $K$-theory, new period spaces in $p$-adic analytic geometry, and new bounds on torsion in singular cohomology), and have made hitherto inaccessible terrains more habitable (e.g., birational geometry in mixed characteristic). This special year intends to bring together a mix of people interested in various facets of the subject, with an eye towards sharing ideas and questions across fields.
The course is an introduction to some of the newest approaches to non-archimedean analytic geometry including:- Huber's adic spaces;- Raynaud's formal schemes and blow-ups;- Clausen-Scholze's analytic spaces.We will focus on specific examples arising from algebraic geometry, Scholze's tilting equivalence of perfectoid spaces and the Fargues-Fontaine curve.We will also show how to define (motivic) homotopy equivalences in this setting, with the aim of defining a relative de Rham cohomology for adic spaces over $\mathbbQ_p$ and a relative rigid cohomology for schemes o
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