Scholze analytic geometry. The topics covered are as follows: 1.

Scholze analytic geometry I called it algebraic geometry because that's how Scholze seems to refer to it at the beginning of his course notes. Clausen and P. On topological cyclic homology (with Thomas Nikolaus), Acta Math 221 (2018), no. Some of the inter-related themes to be explored in this workshop include: the proof of new cases of Langlands reciprocity, linking Galois representations and automorphic forms; breakthroughs in understanding the geometry of Shimura varieties (particularly their analytic geometry over p-adic fields); the growing role of moduli spaces of Galois of Clausen and Scholze. But I wanted to elaborate: Faltings regards the almost purity theorem as an analogue of Zariski-Nagata purity. Peter Scholze Abstract. Berkovich analytic space. We then make detailed discussion on those significant mo-tivic cohomology theories, around perspectives for instance on: Ku¨nneth theorem, 6-functor formalism and so on. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. We choose to work with Huber’s language of adic spaces, However, in this talk, we are interested primarily in examples that go beyond the purview of the Fargues-Scholze heuristic. 12]). References. 28. Lecture I: Introduction Mumford writes in Curves and their Jacobians: “[Algebraic geometry] seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. 3) on preservation of good filtrations ("Donkin subgroups"). He has been a professor at the University of Bonn since 2012 and director at the Max Planck Institute for Mathematics since 2018. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs 33, American Mathematical Society, Providence, 1990. Scholze Real local Langlands as geometric Langlands on the twistor-P1 Fargues’ program for geometrizing the classical local Langlands correspondence through be very helpful, but major aspects of analytic geometry will be recalled in Talk 3 and Talk 4. ] Exactly half a year ago I wrote the Liquid Tensor Experiment blog post, challenging the formalization of a difficult foundational theorem from my Analytic Geometry lecture notes on joint Useful links: 1) Condensed math: [CS] Lectures on Condensed Mathematics (aka the first condensed lecture notes), D. We deduce the weight-monodromy conjecture in certain cases by reduction to equal characteristic. published on October 25, 2023. For a formal scheme \(\mathfrak{X}\) over a height \(1\) valuation ring \(\mathcal{O}\), we construct a Abstract: Berkeley Lectures on p-adic Geometry presents an important breakthrough in arithmetic geometry. Scholze [A1] Pseudocoherent and Perfect Complexes and Vector Bundles on Analytic Adic Spaces, G. Collaborators: Piotr Achinger and Marcin Lara Journal/Status: Compositio Mathematica Tags: Rigid analytic geometry, covering space theory, de Jong fundamental group, pro-étale fundamental group Description. On cyclotomic spectra and topological cyclic homology: On the local Langlands conjecture: On condensed mathematics and analytic geometry: 2. Here is my CV (Last updated: Dec 2024). The language of condensed mathematics was invented in order to build a nicer framework for analytic geometry, Scholze, Peter, and Jared Weinstein, 'Examples of adic spaces', Berkeley Lectures on p-adic Geometry: (AMS-207) The chapter then studies the structure of analytic points. perfectoid field. Place: Y27 H25. 3 Cartier divisors38 In complex geometry, a basic consequence of Hodge theory is that this spectral sequence degenerates at E 1 if X admits a K ahler metric. Recent work of Clausen and Scholze however makes this problem more approachable. 2, 203--409. The ultimate goal of the seminar is to state and prove a version of the Grothendieck–Riemann–Roch theorem for analytic adic spaces using the machinery of condensed mathematics. Geometry Peter Scholze and Jared Weinstein PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2020 "Berkeley Lectures" March 27, 2020 6. Motivated from this, using the approach of Clausen-Scholze of analytic geometry, one can construct different incarnations of the Rham stack that encode different theories of analytic D-modules. [4] B. Current projects involve studying the cohomology of p-adic symmetric domains and the varieties uniformized by them. It replaces the concept of a In this paper, we generalize the result of Mathew to solid quasi-coherent complexes on rigid analytic Expand. Highly We are proud to announce that as of 15:46:13 (EST) on Thursday, July 14 2022 the Liquid Tensor Experiment has been completed. Related concepts. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. global analytic geometry. MATH Google Scholar S. Finally, we use these results to prove the Poincare Duality Theorem for F_p-etale cohomology groups of a smooth proper rigid space over a p-adic field K. This will be discussed further in Section 5. p-adic cohomology. Opening Hausdorff Chair with deadline March 15, 2025. Contribute to PeterScholze/Analytic development by creating an account on GitHub. We will focus on specific examples arising from algebraic geometry, Scholze's tilting equivalence of perfectoid spaces and the Fargues-Fontaine curve. We show Poincaré We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings’ almost purity theorem, and for which there is a natural tilting operation which exchanges characteristic 0 and characteristic p. The topics covered are as follows: By Peter Scholze . Non-Archimedean Analysis: a Systematic Approach to Rigid Analytic Geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 261 (Springer, Berlin, 1984). The proof relies on a novel notion of generic points in rigid analytic geometry which are well-adapted to "spreading out" arguments, in analogy with the use of generic points in scheme theory. Scheduled as part of. perfectoid space. Condensed Mathematics and Complex Geometry, lecture notes for course SS 22. . Office Assistant for the Hausdorff Center for Mathematics (HCM) Angkana Rüland receives Gottfried Wilhelm Leibniz Prize 2025. According to some, [who?] the theory aims to unify various mathematical subfields, including topology, complex geometry, and algebraic $\begingroup$ I just learned about your and Dustin Clausen's work on analytic geometry; it would be an understatement to say that it is amazing! Would it be ok to ask a question about it? From what I understand, this version of analytic geometry includes many other theories as special cases, such as smooth and complex manifolds as well as Berkovich geometry but with application to “analytic” situations, e. We will also Semantic Scholar extracted view of "Lectures on Condensed Mathematics" by P. v3: improved discussion of spectral action with integral coefficients (now also applying in the usual Betti setting), including new results (Theorem VIII. of proof used to prove the corresponding result in p-adic geometry (known as \Tate acyclicity"). Anal Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. 125x9. Zuletzt geändert: September 2022, Peter Scholze. There I was working with Peter Scholze on new foundations for analytic geometry, says •The three sets of lecture notes by Clausen and Scholze: “Condensed Mathematics” [2], “Analytic Geometry” [3], and “Complex Geometry” [1]. Precisely, we prove Artin vanishing in rigid analytic geometry @article{Hansen2017ArtinVI, title={Artin vanishing in rigid analytic geometry}, author= P. Talks 10-11: Towards liquid vector spaces Give the example of an extension of Banach spaces that is not Banach (Corollary 5. non-archimedean analytic geometry. Part of our goal is to develop foundations for analytic geometry that treat archimedean and non-archimedean geometry on equal grounds; and we will proceed by making archimedean geometry more similar to non-archimedean geometry. arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 3. v I want to introduce myself to p-adic geometry. Scholze, “The pro-étale topology for schemes” , This gives a new proof of the finiteness result for ́etale cohomology of proper rigid spaces obtained before in the work of Peter Scholze "p-adic Hodge Theory For Rigid-Analytic Varities". In complex geometry, a basic consequence of Hodge theory is that this spectral sequence degenerates at E 1 if X admits a K ahler metric. mostly been explored in the context of analytic geometry: using condensed sets, a new notion of analytic space can be de ned, which simultaneously generalises scheme theory, complex analytic geometry, rigid analytic geometry, and real manifolds, and simpli es and uni es some imoprtant thoerems in these subjects (see [Sch19b], [Sch19a]). The main prerequisite, in addition to uency with scheme-theoretic algebraic ge-ometry, is a solid command of classical rigid-analytic geometry (such as the We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings’ almost purity theorem, and for which there is a natural tilting operation which exchanges characteristic 0 and characteristic p. It says an analytic stack is a sheaf $\ Algebraic geometry and analytic geometry are closely related (witness GAGA). Analytic stacks. Preface These are lectures notes for a course on analytic geometry taught in the winter term 2019/ at the University of Bonn. But the gluing is not so well worked out. Building on his discovery of perfectoid spaces, the author introduced the concept of “diamonds,” which are to perfectoid spaces what algebraic spaces NON-ARCHIMEDEAN or rigid-analytic geometry is an analog of complex analytic geom-etry over non-Archimedean fields, such as the field of p-adic numbers Qp or the field it has become even more prominent with the work of Scholze and Kedlaya in p-adic Hodge theory, as well as the non-Archimedean approach to mirror symmetry pro-posed by The analytic de Rham stack is a new construction in analytic geometry (after Clausen and Scholze) whose theory of quasi-coherent sheaves encodes a notion of p-adic D-modules, but that has the virtue that it can be defined even under lack of differentials (eg. Building on his discovery of perfectoid spaces, Scholze introduced the concept of “diamonds,” which We construct the moduli space of $p$-adic representations of a profinite group of topologically finite presentation $G$ as a non-archimedean stack. Yichao Tian: Introduction to Rigid Geometry (lecture notes for an Advanced Topics in Algebraic Geometry course) (if you need this, I have a copy) Yichao Tian: Intersection Theory (lecture notes for a Selected Topics in Algebraic Geometry course) Algebraic Geometry I (archived link) as taught by Peter Scholze, notes by Jack Davies The purpose of this course is to propose new foundations for analytic geometry. Talk; Speaker: Juan Esteban Rodriguez Camargo . 03261 Berlin, 1984, A systematic approach to rigid analytic geometry. Peter Scholze (German pronunciation: [ˈpeːtɐ ˈʃɔltsə] ⓘ; born 11 December 1987 [2]) is a German mathematician known for his work in arithmetic geometry. It positively answers the question raised by P. Lecture I: Condensed Sets The basic question to be addressed in this course is the following. Here’s the syllabus: The 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 Part of our goal is to develop foundations for analytic geometry that treat archimedean and non- archimedean geometry on equal grounds; and we will proceed by making archimedean The purpose of this course is to propose new foundations for analytic geometry. These are enough to faithfully capture all sequential spaces, for example, which is In the former theory of analytic geometry, classical abelian or triangulated categories of quasi-coherent sheaves are not enough to obtain descent and glue to more general spaces (a reason is the lack of "complete" flatness even for some simple maps such as open immersions of rigid or complex analytic spaces). We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings’ almost purity theorem, and for which there analytic geometry over K. youtube. Advertisement and p-adic analytic geometry. The new theory of analytic geometry of Clausen and Scholze, together with the stacky approach of integral cohomology theories of Bhatt-Lurie and Drinfeld, have been revealing in how to understand geometrically the very mysterious relationship between theories of D-modules and p-adic Hodge theory. One may say that almost geometrical derived methods could be useful to study integral completed cohomology, while usual derived geometric methods are quite well adapted to the study of torsion phenomena in finite The de Rham stack in algebraic geometry is a geometric object that encodes the theory of D-modules in its theory of quasi-coherent sheaves. The starting point is the observation that the traditional way of endowing https://m. Examples. 10: Condensed mathematics, by Tim. 4 [CS20]). Analytic rings. Ana In this seminar we will study Clausen and Scholze’s new theory of “Condensed Mathematics”, following Scholze’s lecture notes. CHITRABHANU CHAUDHURI is theories of real and complex geometry, which are less amenable to algebraic methods. Abstract In this paper we give an interpretation, in terms of derived de Rham complexes, of Scholze's de Rham period Inspired by Bhatt-Scholze [BS19], in this article, we introduce prismatic cohomology for rigid analytic spaces with l. I haven't taken any number theory or representation theory course. Roughly speaking this consists on constructing an object or a morphism "locally" in a fpqc covering of a scheme, verifying that the local constructions form a descent datum, and using the theorem that says that fpqc descent data are effective to Recently Clausen and Scholze have developed a theory of analytic stack to unify different analytic geometries. Lectures on Analytic The purpose of this course is to propose new foundations for analytic geometry. Scholze in Fall 2014 at UC Berkeley. V. The blueprint for the project can be found here and the formalization What is, in abstract analytic geometry (I mean, for example, in Berkovic spaces), See Clausen–Scholze's Condensed Mathematics and Complex Geometry. In 2014, leading mathematician Peter Scholze delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. 25/01/2021. Ann . I've read chapter 2 and 3 of Hartshorne. Algebraic (or rather analytic) geometry over topological elds Ksuch as R or Q p. 03708) Peter Scholze, Jared Weinstein, Berkeley lectures on p p-adic geometry, pdf. SEMINARS: NUMBER THEORY. Fall 2021 — University of Zürich. In particular, in Lecture 13, they discuss (purely "algebraic" definition Inspired by Bhatt-Scholze [BS19], in this article, we introduce prismatic cohomology for rigid analytic spaces with l. 500. I am also interested in and apply methods from Clausen-Scholze’s condensed mathematics. Reason 2: % 17:06 Of all the above, the only rich theory allowing both Archimedean and non-Archimedean geometry is Berkovich's theory. When: February 3, 2021. Bhatt and P. Light condensed abelian groups. Scholze in [Sch13a]. 11811/10125, urn: https://nbn-resolving. Scholze [CS2] Lectures on Analytic Geometry (aka the second condensed lecture notes), D. Rigid spaces, Math. Date: Tue, 20/06/2023 using analytic geometry of Clausen and Scholze, I will explain how to construct similar de Rham stacks for rigid spaces (and some relatives) that encode the theory of The purpose of this course is to propose new foundations for analytic geometry. I am working in the field of arithmetic geometry, with my main interest lying in the geometric Langlands program. As taught in school books, analytic geometry can be explainedmore simply: it is concerned with defining Analytic Stacks . Programmvorschlag: J. This assumption is not necessary in p-adic geometry: Theorem 2. The topics covered are as follows: 1. Speaker: Peter Scholze - University of Bonn We will outline a definition of analytic spaces that relates to complex- or rigid-analytic varieties in the same way that schemes relate to algebraic varieties over a field. algebraic-geometry; complex-geometry; non-archimedean-fields; condensed-mathematics; Wojowu. Recently Clausen and Scholze have developed a theory of analytic stack to unify different analytic geometries. Moreover it is better suited to face the topological issues that one encounters in these theories. We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of 𝐂 p. A year and a half after the challenge was posed by Peter Scholze we have finally formally verified the main theorem of liquid vector spaces using the Lean proof assistant. In December 2020, Peter Scholze posed a challenge to formally verify the The purpose of this course is to propose new foundations for analytic geometry. Étale cohomology of diamonds, Preprint, 2018. We will focus on the first non-trivial case of the infinite level modular curve, where we can describe the main points without assuming too much background with perfectoid rings or p-adic geometry. This semester we plan to discuss the new approach to non-archimedean analytic geometry due to Clausen–Scholze. Peter Many contexts are expected to be sheafified, such as over Scholze's pro-étale sites of the considered analytic spaces by using perfectoids or the quasisyntomic sites by using quasiregular semiperfectoids as in the work of Bhatt-Morrow-Scholze and Bhatt-Scholze. 008 at the University, while Thursdays, 12:15 -- 14:00, I will hold the May 2019 Peter Scholze 5. Is there a nontrivial way to consider products of archimedean and non-archimedean spaces in the context of Clausen–Scholze's analytic geometry? Context: Last week during a conference in Essen (School ag. It also clarifies the relations between analytic rings and Tate rings. Rapoport-Zink space. 2024) Vanishing and comparison theorems in rigid analytic geometry ( pdf ) Compositio Math. p-adic physics. Institute for Advanced Study 1 Einstein Drive Princeton, New Jersey 08540 USA This is closely related to Ayoub's theory of rigid-analytic motives, but works uniformly in the archimedean and nonarchimedean setting. p-ADIC GEOMETRY PETER SCHOLZE Abstract. Introduction. @phdthesis{handle:20. for perfectoid spaces or Fargues-Fontaine curves). These are notes of a seminar held in Columbia university during the Spring of 2024 about the new theory of analytic stacks of Clausen and Scholze. prismatic cohomology. Via the methods of condensed mathematics [4], one can construct theories of analytic and complex geometry [5], [6] with well-behaved categorical properties and so which can be Peter Scholze, Max Planck Institute for Mathematics and University of Bonn . 0. , Scholze, Peter Homepage of Peter Scholze Lecture (Summer 2022): Condensed Mathematics and Complex Geometry This course is joint with Dustin Clausen (University of Copenhagen) and will be held hybrid: On Tuesdays, 12:15 -- 14:00, Clausen will hold the lecture in Copenhagen and we will watch it in Room N0. Time: Fridays 15:00-17:00. Visit Stack Exchange that underlies p-adic Hodge theory. 1 Adic morphisms35 5. [This is a guest post by Peter Scholze. It takes values in a mixed-characteristic analogue of Dieudonné modules, which was previously defined by Fargues as a version of Breuil–Kisin modules. From Hensel to Berkovich and beyond, talk notes, June 2012 . 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. Comments welcome! This book presents an important breakthrough in arithmetic geometry. Peter Scholze, p p-adic geometry, ICM report (arXiv:1712. Scholze in his paper ``p-adic Hodge Theory for Rigid-analytic varieties''. Condensed Mathematics and Analytic Geometry. formal schemes, adic spaces, or More abstractly, one could say following Scholze that it should be “compactifiable”; more abstractly yet, one can notice that this really just requires isolating two classes of maps Iand P In the recent lecture series run jointly from IHÉS and Bonn, Clausen and Scholze have reworked—again—their foundations of geometry to focus attention on not arbitrary condensed sets and solid modules and so on, but the much smaller class of light condensed sets and so on. 3 The pro-´etale site of an analytic adic space He is known for his work on algebraic K-theory, on connections between homotopy theory and arithmetic, and more recently and jointly with Peter Scholze, on the development of condensed mathematics and the attendant approach to analytic geometry. What should I learn before going into the articles of Scholze or Bhatt? If you link some books or articles, I would much appreciate that. Ansch utz, P. 6 CONDENSED MATHEMATICS 1. In particular, any variety Xover Khas an associated adic space Xad over K, which in Stack Exchange Network. For any proper smooth rigid-analytic space Xover C, the Hodge-to-de Rham spectral sequence isomorphism has to be of an analytic nature, and we have to use some kind of rigid-analytic geometry over K. , Vol. pro-étale morphism. g. Condensed mathematics o ers a new way to develop a theory of analytic geometry, which uni es both the geometry of complex analytic spaces and the p-adic theories of rigid/Berkovich/adic spaces. 01:38:10. No originality is claimed. We deduce the weight-monodromy conjecture in certain cases by reduction to equal characteristic. Actually I do not know many details about it. For an article (in German) about them and the topic of the course, see here. 8], [BMS16, Theorem 13. Program. Anal Dustin Clausen and Peter Scholze are giving a course together this fall on Analytic Stacks, with Clausen lecturing at the IHES, Scholze from Bonn. Abel in Bonn: Abel Symposium 2025. We give We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings’ almost purity theorem, and for which there is a natural tilting operation which exchanges characteristic 0 and characteristic p. B. Perfectoid spaces, rigid-analytic geometry, almost mathematics, p-adic Hodge theory, Shimura varieties, Langlands program. org/urn:nbn:de:hbz:5-67399, author = {{Lucas Mann}}, title = {A p-Adic 6-Functor Formalism in Rigid V5A2 - Analytic Geometry Fr, 12:00 -- 14:00, Zeichensaal Arithmetische Geometrie Oberseminar (ARGOS): Prismatic Dieudonne Theory Zuletzt geändert: März 2022, Peter Scholze. I completed my PhD at the University of Bonn under the supervision of Peter Scholze. As an application, we develop a six functor formalism It positively answers the question raised by P. 1. analytic models for schemes over Z may be considered as “Arakelov type” models. Using this new setting for global analytic geometry, we will define in Section 6 various cohomological invariants, as étale, analytic motivic cohomology, and global analytic K-theory. Morrow and P. Skip to main content. I want to study the theory of Condensed Mathematics and Analytic Geometry by Scholze and Clausen. Affiliation: MPIM . Bosch and W. p-adic geometry, Proceedings of the ICM 2018. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze’s perfectoid spaces. Geometrization of the local Langlands correspondence , lecture notes and videos. At a few points, we have expanded slightly on the material, in particular so as to provide a full construction of local Shimura varieties and general moduli spaces of shtukas, Peter Scholze (Max Planck Institute for Mathematics and University of Bonn)Conference on Homotopy Theory with Applications to Arithmetic and GeometryJune 27t Peter Scholze proposes to formalize the proof of a major result in the Clausen–Scholze approach to rigid analytic geometry via condensed sets. I. p-adic Hodge theory. @article{Scholze2017EtaleCO, title={Etale cohomology of diamonds}, author={Peter Scholze}, journal={arXiv: Algebraic Geometry}, year ={2017 a new approach to relative p-adic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry in the style of both Berkovich and Huber. We don’t have a definition of Kähler rigid-analytic varieties, but this examples hould not be Kähler. Analytic Stacks (3/29) By We want to accommodate all the examples. Keywords: adic spaces, Berkeley Lectures on p-adic Geometry presents an important breakthrough in arithmetic geometry. shtuka. i singularities, with coefficients over Fontaine’s de Rham period ring B dR . Skip to search form Skip is a new way of studying the interplay between algebra and geometry. Peter Scholze Notes by Tony Feng 1 Classical Hodge Theory Dividing by qZ is bad in algebraic geometry because this is not a proper discontinuous action, but it is in rigid analytic geometry. c. Scholze. The seminar is inspired from the Lecture Series of Analytic Stacks, all results are due to Clausen and Scholze, and any mistake or misconception is completely due to the author. Last arXivLabs: experimental projects with community collaborators. i singularities we record some foundational results on adic geometry that seem to be Institute for Advanced Study 1 Einstein Drive Princeton, New Jersey 08540 USA PETER SCHOLZE Abstract. 2 Analytic adic spaces36 5. 156 Issue 2, pp. We prove duality via constructing Faltings' trace map relating Poincaré Duality on the generic fiber to (almost) Mod-p Poincaré Duality in p-adic Analytic Geometry Zavyalov, Bogdan; Abstract. Dustin Clausen did his undergraduate studies at Harvard in Rigid-Analytic Geometry Lucas Mann June 7, 2022 We develop a full 6-functor formalism for p-torsion étale sheaves in rigid-analytic geometry. A common technique for constructing objects (sheaves) and morphisms in algebraic geometry is faithfully flat descent. Building on his discovery of perfectoid spaces, Scholze introduced the concept of “diamonds,” which are to The course consists in an introduction to non-archimedean arithmetic geometry using the language of adic spaces, formal schemes and condensed rings. Exposition includes. 12 Rigid spaces as analytic stacks and GAGA (Lectures 15 and 19 of AS) We show that rigid spaces can be constructed as analytic stacks. More generally, it Is there a nontrivial way to consider products of archimedean and non-archimedean spaces in the context of Clausen–Scholze's analytic geometry?. Mathematics Subject Classi cation (2010). News Mourning Tim Lichtnau. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions). Thanks in advance. Lütkebohmert, Formal and rigid geometry. We aim for a self-contained treatment, not relying on previous work on algebraic or analytic motives. p-adic homotopy. For this course, only the simplest case where Xis a (geometric) point is relevant. We prove a generic smoothness result in rigid analytic geometry over a characteristic zero nonarchimedean field. In-troduce the notion of p-locally convex vector space, the analytic ring of p-liquid R-vector spaces and Vector Bundles on Analytic Adic Spaces Grigory Andreychev Abstract Using the new approach to analytic geometry developed by Clausen and Scholze by means of condensed mathematics, we prove that for every affinoid analytic adic space X, pseudocoherent complexes, perfect complexes, and finite projective modules over O X(X)form a stack with respect The purpose of this course is to propose new foundations for analytic geometry. Both approaches may be extended to the analytic setting using overconvergent derived analytic spaces (see global analytic geometry). local Langlands correspondence. 3 Analytic points32 Lecture 5: Complements on adic spaces35 5. Scholze's work with perfectoid spaces and has novel ways to address this problem by, in a way, making p-adic geometry more analytic. Authors Bogdan Zavyalov Princeton University, Princeton, NJ, and Institute for Advanced Study, Princeton, NJ Analytic geometry. Introduction -ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIES - Volume 1. Applying the new theory of analytic stacks of Clausen and Scholze we introduce a general notion of derived Tate adic spaces. 07738: The analytic de Rham stack in rigid geometry. This book discusses the theory of perfectoid spaces and their applications and aims to give an introduction to Scholze’s theory. 4. 5k; asked Sep 10, 2022 at 13:17. Andreychev 2) K-theory: 4:30 PM – 5:30 PM Tomer Schlank, p-adic Analytic Geometry and Chromatic Homotopy Theory Wednesday, March 13, 2024: 10:00 AM – 11:00 AM Peter Scholze, Analytic Prismatization 11:00 AM – 12:00 PM Coffee/Tea in Rubenstein Commons Cafe This is really just an elaboration of Emerton's comment: You should read Mark Kisins' review of Faltings's paper "Almost etale extensions". Talk: Overview (9. Google Scholar. Subsequent work by Scholze has demonstrated without a doubt that perfectoid spaces are a powerful new tool across many aspects of algebraic number theory. Talks: 29. p-adic geometry, d'apres Peter Scholze Notes from an informal lecture series I gave at Columbia in Spring 2015, on Scholze's Berkeley course and related material. One of the main conceptual I don't pretend to have anything more than a superficial understanding of condensed mathematics, but Scholze's lecture notes (on condensed mathematics and analytic geometry) are so clearly written that you can get some sense of the main idea just from studying the first few pages. More concretely, we use the recently developed condensed mathematics by Clausen–Scholze to associate to every small v-stack (e. For any proper smooth rigid-analytic space Xover C, the Hodge-to-de Rham spectral sequence This is a survey article over some of the work of Peter Scholze for the Jahresbericht der DMV. We survey the theory of perfectoid spaces and its applications. After my postdoc here at UCPH, I went to Bonn for two years, first at the University of Bonn and then at the Max Planck Institute for Mathematics. However, various foundational problems remain, for example: We give the formal definition of analytic stacks using the!-topology, then we give examples of analytic stacks: light condensed anima, derived schemes, rigid spaces, complex and real analytic spaces,etc. Schedule of Talks. ´ •A detailed computation of a fundamental counterexample in the theory of liquid vector Specialization for the Pro-étale Fundamental Group. analytic geometry and does not presume a deep knowledge of algebraic We establish several new properties of the $p$-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. Mathematics. The de Rham stack for rigid analytic varieties. Also, I only ever really learned how to take derivatives so if I see someone computing an Ext group in order to understand a space, the only thing I know to call this is "algebraic geometry. We prove a rigid analytic analogue of the Artin vanishing theorem. $\begingroup$ Perhaps you'll want to first take a look at some more classical approaches to analytic geometry (rigid geometry, formal schemes), the applications that they have, Due to the sheer amount of extra structure in p-adic geometry, it is hard to find a cohomology that is both computable that also does not forget "too much" information. You'd like a general theory of analytic spaces which can be specialized to whicheever context you might be interested in. Context: Last week during a conference in Essen (School on Arithmetic Geometry) Peter Scholze has given a talk about his (joint with Dustin Clausen) work on analytic geometry. He has been called one of the leading mathematicians in the world. •The master thesis of Dagur Asgeirsson. Conference on Homotopy Theory with Applications to Arithmetic and Geometry. 1 ([Sch13a, Corollary 1. com/watch?v=B076k82McTc Motivated from the theory of prismatic cohomology of Bhatt-Scholze, and its geometrization due to Drinfeld and Bhatt-Lurie after Scholze’s recent theory of Berkovich motives, Scholze’s theory of small E-stacks and Clausen-Scholze’s analytic stacks. The purpose of this course is to propose new foundations for analytic geometry. 4. We discuss recent developments in p-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for “compact p-adic manifolds” over new period maps on moduli spaces of abelian varieties to applications to the local and global Langlands Condensed mathematics is a theory developed by Dustin Clausen and Peter Scholze which replaces a topological space by a certain sheaf of sets, in order to solve some technical problems of doing homological algebra on topological groups. p-adic geometry Preface This is a revised version of the lecture notes for the course on p-adic geometry given by P. Talks 1. 12591: Pseudocoherent and Perfect Complexes and Vector Bundles on Analytic Adic Spaces Abstract page for arXiv paper 2401. Posted in . 25 4. Primary: 14G22, 11F80 Secondary: 14G20, 14C30, 14L05, 14G35, 11F03 Keywords. We will provide some fundamental examples such as Tate's analytic varieties, Scholze's perfectoid spaces and the Fargues-Fontaine curve. Applying the theory to discrete fields, one still recovers the etale version of Voevodsky's theory. Comments: 356 pages, v4: accepted version. October 2019, Peter Scholze. Lectures on Analytic Geometry Peter Scholze (all results joint with Dustin Clausen) Analytic Geometry. Office Assistant for the Hausdorff Center for Mathematics (HCM) At the IHES and the Max Planck Institute, the Clausen-Scholze joint course on analytic stacks has just ended. 2. Lectures will be given by Dustin Clausen at IHES and Peter Scholze at MPI, and broadcast live at the other location. In collaboration with Peter Scholze he has recently developed condensed mathematics, a new theory of analytic geometry, combining algebra and topology. This theory, a version of which was also discovered independently by Barwick and Haine and called pyknotic sets, has two localizations, giving the theory of solid and liquid condensed sets, aimed at the study of We compute the Balmer spectra of compact objects of tensor triangulated categories whose objects are filtered or graded objects of (or sheaves valued in) another tensor triangulated category. The first course, taught by Peter Scholze, contains his recent results dealing with the local Langlands conjecture. What they’re working on provides some We will focus on the following two recent papers of Dustin Clausen and Peter Scholze: Lectures on Condensed Mathematics Lectures on Analytic Geometry An alternative approach has been proposed in a paper by Clark Barwick and Peter Haine. rigid-analytic variety) X with pseudouniformizer π an ∞-category Fair point. Joint with Abstract page for arXiv paper 2105. Related entries. Dustin Clausen, Peter Scholze, Masterclass in condensed mathematics, YouTube playlist, website (including pdf notes) Peter Scholze, Lectures on analytic geometry, pdf. Dustin Clausen, Peter Scholze, Condensed Mathematics and Complex Geometry, pdf. 1 During the past few years, Dustin Clausen has worked in collaboration with mathematician and Fields medalist Peter Scholze, on developing an important new, general theory of analytic geometry, combining algebra and Berkeley lectures on p-adic geometry (with Jared Weinstein), Annals of Math Studies 207. arXivLabs: experimental projects with community collaborators. " Peter Scholze, Lectures on analytic geometry, pdf. Introduction In August 2018, Scholze was awarded a Fields medal “for transforming arithmetic algebraic geometry over p-adic fields through his introduction of 5. Lukas Brantner: Overview 01/02/2021. What is a good roadmap for my background. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Brinon, O. We choose to work with Huber’s language of adic spaces, which reinterprets rigid-analytic varieties as certain locally ringed topological spaces. It says an analytic stack is a sheaf $\mathrm{AnRing}\rightarrow \mathrm{Ani}$ from the category of analytic rings to the category of anima (spaces) satisfying $!$-descent for $!$-hypercovers. Michael Harris, The perfectoid concept: Test case for an absent theory ; The concept is due to. Indeed, unlike in classical topologies such as the Zariski or etale topology, sheaves on the pro- etale site of a lecture notes for Analytic Geometry course. Scholze, Topological Hochschild homology and integral \(p\)-adic Hodge theory, available at arXiv:1802. 5 6 ANALYTIC GEOMETRY. We will discuss some of our joint work with Dustin Clausen on Analytic Geometry, based on Condensed Mathematics. Bhatt, M. Research. 299-324. gzwx wtqot ljpr hppbn bjrgtsde mralh kcap pxdfy aarlic hnqa