2023 IHES Summer School - Recent Advances in Algebraic K-theory

10 juil. 2023, 08:30 → 21 juil. 2023, 19:00 Europe/Paris

Marilyn and James Simons Conference Center (IHES)

2023 IHES SUMMER SCHOOL

Organizing Committee: Benjamin Antieau (Northwestern University), Lars Hesselholt (University of Copenhagen / Nagoya University), and Matthew Morrow (CNRS and Université Paris-Saclay)

Scientific Committee: Bhargav Bhatt (IAS and Princeton University / University of Michigan), Wiesia Niziol (CNRS and Sorbonne Université), and Akhil Mathew (University of Chicago) 

The Summer School will be held at the Institut des Hautes Etudes Scientifiques (IHES) from July 10 to 21, 2023. IHES is located in Bures-sur-Yvette, south of Paris (40 minutes by train from Paris) - Access map

This school is open to everybody but intended primarily for young participants, including Ph.D. students and postdoctoral fellows. 

Please note that there won't be remote transmission through Zoom but mini-courses and talks will be filmed and posted on the IHES YouTube channel in the following days.

Application is open until February 15, 2023. 

In the style of an Oberwolfach Arbeitsgemeinschaft, ten talks will be given by postdoctoral participants on the topic of syntomic and étale motivic cohomology. Once the detailed list of talks is available, postdoctoral applicants will be contacted to ask which talk they would be willing to give.

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2023 IHES Summer School - Recent Advances in Algebraic $K$-theory

The last few years have witnessed an explosion of progress in algebraic $K$-theory. Derived algebraic geometry and non-commutative methods have been refined into powerful tools, especially through the theory of localizing invariants. Trace methods have brought $K$-theory and topological cyclic homology closer together than ever before. Perfectoid techniques mean that $K$-theory benefits from the recent progress in $p$-adic cohomology, such as prismatic cohomology. Condensed mathematics provides at long last a uniform approach to the $K$-theory of topological rings. Geometric foundations for motivic stable homotopy theory have been laid and new motivic filtrations have been unearthed.

The goal of the Summer School will be to help bring the participants up to date on these exciting developments, via research lectures, mini-courses, and an Arbeitsgemeinschaft on the topic of syntomic and étale motivic cohomology.

MINI-COURSES:

• Johannes Anschutz (University of Bonn) and Arthur-César Le Bras (CNRS and Université de Strasbourg)
• Dustin Clausen (University of Copenhagen and IHES)
• Elden Elmanto (Harvard University)
• Ryomei Iwasa (Université Paris-Saclay)
• Georg Tamme (University of Mainz)

SPEAKERS:

• Kęstutis ČESNAVIČIUS (CNRS and Université Paris-Saclay)
• Shane KELLY (University of Tokyo)
• Moritz KERZ (University of Regensburg)
• Hana Jia KONG (Institute for Advanced Study)
• Achim KRAUSE (University of Münster)
• Thomas NIKOLAUS (University of Münster)
• Arpon RAKSIT (Massachusetts Institute of Technology)
• Charanya RAVI (Indian Statistical Institute, Bangalore)
• Kirsten WICKELGREN (Duke University)
• Maria YAKERSON (CNRS and Sorbonne Université)

 

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This is an IHES Summer School organized in partnership with the Clay Mathematical Institute and in part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101001474).

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lundi 10 juillet

09:00 → 09:30 Welcome coffee and registration

09:30 → 10:45 Arbeitsgemeinschaft 1: Introduction

An introduction to the Arbeitsgemeinschaft on syntomic cohomology, and an overview of parts of the theory in characteristic p.

Orateur: Matthew Morrow (CNRS)

10:45 → 11:15 Coffee break / discussion

11:15 → 12:30 Localizing Invariants and Algebraic K-theory (1/3)

It was a fundamental insight by Thomason (building on work of Waldhausen) that algebraic K-theory of a ring or scheme could be defined via the category of perfect complexes and that K-theory sends a Verdier quotient sequence of such categories to a fiber sequence of spectra. In modern terms: K-theory is a localizing invariant. This in particular yields descent properties, e.g. Nisnevich descent, for K-theory or more generally any localizing invariant. I plan to give an introduction to these by now classical topics and discuss recent developments and applications, in particular for algebraic K-theory.

Orateur: Prof. Georg Tamme (University of Mainz)

Vidéo

12:30 → 14:30 Lunch break (buffet at IHES)

14:30 → 16:00 Introduction to Prismatic Cohomology (1/4)

This series of four lectures will offer an introduction to prismatic cohomology as developed by Bhatt and Scholze. More concretely, we plan to cover:
1) Prismatic cohomology in char 0 and characteristic p
2) General results on (derived) prismatic cohomology
3) Prismatization as a tool for understanding prismatic cohomology
4) The prismatic logarithm

Orateur: Prof. Johannes Anschütz (University of Bonn)

Vidéo

16:00 → 16:30 Coffee break / discussion

16:30 → 17:30 A Homotopical Approach to Crystalline Cohomology

A naive approach to crystalline cohomology is to lift a smooth ring to characteristic zero and then to take de Rham cohomology. However, due to the non-canonical local nature of this process, it isn't easy to glue in order to define crystalline cohomology globally by this method. We consider a variant of this approach by taking higher homotopies into account, making the lift canonical and gluing easy. (Joint work with G. Tamme.)

Orateur: Prof. Moritz Kerz (University of Regensburg)

Vidéo

mardi 11 juillet

08:45 → 09:15 Welcome coffee

09:15 → 10:45 Introduction to Prismatic Cohomology (2/4)

This series of four lectures will offer an introduction to prismatic cohomology as developed by Bhatt and Scholze. More concretely, we plan to cover:
1) Prismatic cohomology in char 0 and characteristic p
2) General results on (derived) prismatic cohomology
3) Prismatization as a tool for understanding prismatic cohomology
4) The prismatic logarithm

Orateur: Prof. Johannes Anschütz (University of Bonn)

Vidéo

10:45 → 11:15 Coffee break / discussion

11:15 → 12:30 Localizing Invariants and Algebraic K-theory (2/3)

It was a fundamental insight by Thomason (building on work of Waldhausen) that algebraic K-theory of a ring or scheme could be defined via the category of perfect complexes and that K-theory sends a Verdier quotient sequence of such categories to a fiber sequence of spectra. In modern terms: K-theory is a localizing invariant. This in particular yields descent properties, e.g. Nisnevich descent, for K-theory or more generally any localizing invariant. I plan to give an introduction to these by now classical topics and discuss recent developments and applications, in particular for algebraic K-theory.

Orateur: Prof. Georg Tamme (University of Mainz)

Vidéo

12:30 → 14:30 Lunch break

14:30 → 15:45 [Videoconference] On the Motivic Cohomology of Schemes (1/3)

We outline the theory of motivic cohomology of general equicharacteristic schemes, as developed jointly with Matthew Morrow. Roughly, the lectures will be divided as follows:
Lecture 1: cdh and A^1-invariant motivic cohomology. I will first give a general, leisurely introduction to the cdh topology and some of its applications to algebraic geometry and K-theory.
Lecture 2: the construction of the motivic filtration on K-theory. I will then explain how to construct a motivic filtration on K-theory by gluing together the theory of syntomic cohomology and A^1-invariant/cdh motivic cohomology. Some of the results presented here are joint with Tom Bachmann and Matthew Morrow.
Lecture 3: a sampler of motivic cohomology. I will then give some features of the resulting theory of motivic cohomology. Topics include an extension of the Nesterenko-Suslin isomorphism (with Milnor K-theory), a motivic refinement of Weibel's vanishing conjecture, and results on zero cycles.

Orateur: Prof. Elden Elmanto (University of Toronto)

Vidéo

15:45 → 16:30 Coffee break / discussion

16:30 → 17:30 Structures and Computations in the Motivic Stable Homotopy Categories

A fundamental question in classical stable homotopy theory is to understand the stable homotopy groups of the spheres. A relatively new method is via the motivic approach. Motivic stable homotopy theory has an algebro-geometric root and closely connects to questions in number theory. Besides, it relates to the classical and the equivariant theories. The motivic category has good properties and allows different computational tools. I will talk about some of these properties and computations and will show how it relates to the classical and equivariant categories. This is joint work with Tom Bachmann, Guozhen Wang, and Zhouli Xu.

Orateur: Prof. Hana Jia Kong (Institute for Advanced Study)

mercredi 12 juillet

08:45 → 09:15 Welcome coffee

09:15 → 10:45 Introduction to Prismatic Cohomology (3/4)

This series of four lectures will offer an introduction to prismatic cohomology as developed by Bhatt and Scholze. More concretely, we plan to cover:
1) Prismatic cohomology in char 0 and characteristic p
2) General results on (derived) prismatic cohomology
3) Prismatization as a tool for understanding prismatic cohomology
4) The prismatic logarithm

Orateur: Prof. Johannes Anschütz (University of Bonn)

Vidéo

10:45 → 11:15 Coffee break / discussion

11:15 → 12:30 Arbeitsgemeinschaft 2

Definition of syntomic cohomology of p-complete (animated) rings, basic properties and examples.

Orateur: Noah Riggenbach (Northwestern University)

12:30 → 14:30 Lunch break

14:30 → 15:45 Localizing Invariants and Algebraic K-theory (3/3)

It was a fundamental insight by Thomason (building on work of Waldhausen) that algebraic K-theory of a ring or scheme could be defined via the category of perfect complexes and that K-theory sends a Verdier quotient sequence of such categories to a fiber sequence of spectra. In modern terms: K-theory is a localizing invariant. This in particular yields descent properties, e.g. Nisnevich descent, for K-theory or more generally any localizing invariant. I plan to give an introduction to these by now classical topics and discuss recent developments and applications, in particular for algebraic K-theory.

Orateur: Prof. Georg Tamme (University of Mainz)

Vidéo

15:45 → 16:30 Coffee break / discussion

16:30 → 17:30 Algebraic K-Theory in Geometric Topology

While algebraic K-theory has many applications and uses in modern algebra and arithmetic, its origins actually lie in geometric topology through Whithead's work on simple homotopy theory. We will review this in modern language, to elucidate the nature of simple homotopy types. This will also lead to parametrized versions of results of West and Chapman through the use of Efimov-K-Theory. The key is a K-theoretic model of assembly maps which is of independent nature and should have many more applications in the furture. This is joint work with A. Bartels and A. Efimov.

Orateur: Prof. Thomas Nikolaus (University of Münster)

Vidéo

jeudi 13 juillet

08:45 → 09:15 Welcome coffee

09:15 → 10:45 Introduction to Prismatic Cohomology (4/4)

This series of four lectures will offer an introduction to prismatic cohomology as developed by Bhatt and Scholze. More concretely, we plan to cover:
1) Prismatic cohomology in char 0 and characteristic p
2) General results on (derived) prismatic cohomology
3) Prismatization as a tool for understanding prismatic cohomology
4) The prismatic logarithm

Orateur: Prof. Johannes Anschütz (University of Bonn)

Vidéo

10:45 → 11:15 Coffee break / discussion

11:15 → 12:30 Arbeitsgemeinschaft 3

Syntomic first Chern class; first part of proof that it is an equivalence after p-completion.

Orateur: Emanuel Reinecke (MPIM Bonn)

12:30 → 14:30 Lunch break

14:30 → 15:45 [Videoconference] On the Motivic Cohomology of Schemes (2/3)

We outline the theory of motivic cohomology of general equicharacteristic schemes, as developed jointly with Matthew Morrow. Roughly, the lectures will be divided as follows:
Lecture 1: cdh and A^1-invariant motivic cohomology. I will first give a general, leisurely introduction to the cdh topology and some of its applications to algebraic geometry and K-theory.
Lecture 2: the construction of the motivic filtration on K-theory. I will then explain how to construct a motivic filtration on K-theory by gluing together the theory of syntomic cohomology and A^1-invariant/cdh motivic cohomology. Some of the results presented here are joint with Tom Bachmann and Matthew Morrow.
Lecture 3: a sampler of motivic cohomology. I will then give some features of the resulting theory of motivic cohomology. Topics include an extension of the Nesterenko-Suslin isomorphism (with Milnor K-theory), a motivic refinement of Weibel's vanishing conjecture, and results on zero cycles.

Orateur: Prof. Elden Elmanto (University of Toronto)

Vidéo

15:45 → 16:30 Coffee break / discussion

16:30 → 17:30 Prismatic Cohomology of Commutative Ring Spectra

I will discuss motivic filtrations on trace invariants of commutative ring spectra, defined in joint work with Jeremy Hahn and Dylan Wilson, and I will report on further work in progress concerning the associated graded objects of these filtrations, which constitute an extension of the theory of prismatic cohomology to the setting of commutative ring spectra.

Orateur: Prof. Arpon Raksit (Massachusetts Institute of Technology)

Vidéo

18:00 → 19:00 Buffet dinner at IHES

vendredi 14 juillet

09:00 → 17:00 July 14 - Public Holiday (French National Day)

lundi 17 juillet

08:45 → 09:15 Welcome coffee

09:15 → 10:30 Arbeitsgemeinschaft 4

Second step of proof that syntomic first Chern class is equivalence after p-completion.

Orateur: Shubhodip Mondal (MPIM Bonn)

10:30 → 11:15 Coffee break / discussion

11:15 → 12:30 [Videoconference] On the Motivic Cohomology of Schemes (3/3)

We outline the theory of motivic cohomology of general equicharacteristic schemes, as developed jointly with Matthew Morrow. Roughly, the lectures will be divided as follows:
Lecture 1: cdh and A^1-invariant motivic cohomology. I will first give a general, leisurely introduction to the cdh topology and some of its applications to algebraic geometry and K-theory.
Lecture 2: the construction of the motivic filtration on K-theory. I will then explain how to construct a motivic filtration on K-theory by gluing together the theory of syntomic cohomology and A^1-invariant/cdh motivic cohomology. Some of the results presented here are joint with Tom Bachmann and Matthew Morrow.
Lecture 3: a sampler of motivic cohomology. I will then give some features of the resulting theory of motivic cohomology. Topics include an extension of the Nesterenko-Suslin isomorphism (with Milnor K-theory), a motivic refinement of Weibel's vanishing conjecture, and results on zero cycles.

Orateur: Prof. Elden Elmanto (University of Toronto)

Vidéo

12:30 → 14:30 Lunch break

14:30 → 15:45 Efimov K-theory (1/3)

The scope of algebraic K-theory has been greatly extended by Alexander Efimov, who showed that it can be defined in a much broader categorical context than previously realized. This extended scope allows for precise connections with topology and analysis. I will give an introduction to Efimov's theory, and then focus on applications to complex geometry, the latter being joint work with Peter Scholze.

Orateur: Prof. Dustin Clausen (University of Copenhagen & IHES)

Vidéo

15:45 → 16:30 Coffee break / Q&A

16:30 → 17:30 The Pro-cdh Topology

This will essentially be the talk I gave in Cambridge last year, except that now all the details have been worked out, and we don't explicitly need ind-schemes anymore.

Orateur: Prof. Shane Kelly (University of Tokyo)

Vidéo

mardi 18 juillet

08:45 → 09:15 Welcome coffee

09:15 → 10:30 Arbeitsgemeinschaft 5

Third step of proof that syntomic first Chern class is equivalence after p-completion.

Orateur: Bogdan Zavyalov (IAS)

10:30 → 11:15 Coffee break / discussion

11:15 → 12:30 Motivic Stable Homotopy Theory (1/3)

In joint work with Toni Annala and Marc Hoyois, we have developed motivic stable homotopy in broader generality than the theory initiated by Voevodsky, so that non-$A^1$-invariant theories can also be captured. I’ll describe this, bearing in mind its connection to algebraic K-theory and p-adic cohomology such as syntomic cohomology. The course is divided roughly into three parts.
Foundations: The goal of this part is to grasp the notion of $P^1$-spectrum, which forms the basic framework of motivic stable homotopy theory.
Techniques: The goal of this part is to understand our main technique, P-homotopy invariance, which allows us to do a homotopy theory in algebraic geometry while keeping the affine line $A^1$ non-contractible.
Applications: In this part, we apply our motivic homotopy theory to algebraic K-theory of arbitrary qcqs schemes, and prove an algebraic analogue of Snaith theorem, which says that K-theory is obtained from the Picard stack by inverting the Bott element.

Orateur: Prof. Ryomei Iwasa (Université Paris-Saclay)

Vidéo

12:30 → 14:30 Lunch break

14:30 → 15:45 Efimov K-theory (2/3)

The scope of algebraic K-theory has been greatly extended by Alexander Efimov, who showed that it can be defined in a much broader categorical context than previously realized. This extended scope allows for precise connections with topology and analysis. I will give an introduction to Efimov's theory, and then focus on applications to complex geometry, the latter being joint work with Peter Scholze.

Orateur: Prof. Dustin Clausen (University of Copenhagen & IHES)

Vidéo

15:45 → 16:30 Coffee break / Q&A

16:30 → 17:30 K-theoretic Localization Theorem

The classical localization theorem due to Borel, Atiyah-Segal, and Quillen says that the equivariant cohomology of a space can be recovered, up to inverting some elements, from the equivariant cohomology of the fixed point subspace. A version of this result for topological K-theory was proved by Segal in 1968. In this talk, we discuss the algebraic analog of this result, due to Thomason for schemes, and its extension to algebraic stacks. We also formulate the Atiyah-Bott, Graber-Pandharipande virtual localization formula for the structure sheaf. This is based on a joint work in progress with Adeel Khan and Hyeonjun Park.

Orateur: Prof. Charanya Ravi (Indian Statistical Institute)

Vidéo

mercredi 19 juillet

08:45 → 09:15 Welcome coffee

09:15 → 10:30 Arbeitsgemeinschaft 6

Construction of syntomic-to-étale comparison maps

Orateur: Guido Bosco (MPIM Bonn)

10:30 → 11:15 Coffee break / discussion

11:15 → 12:30 Motivic Stable Homotopy Theory (2/3)

In joint work with Toni Annala and Marc Hoyois, we have developed motivic stable homotopy in broader generality than the theory initiated by Voevodsky, so that non-$A^1$-invariant theories can also be captured. I’ll describe this, bearing in mind its connection to algebraic K-theory and p-adic cohomology such as syntomic cohomology. The course is divided roughly into three parts.
Foundations: The goal of this part is to grasp the notion of $P^1$-spectrum, which forms the basic framework of motivic stable homotopy theory.
Techniques: The goal of this part is to understand our main technique, P-homotopy invariance, which allows us to do a homotopy theory in algebraic geometry while keeping the affine line $A^1$ non-contractible.
Applications: In this part, we apply our motivic homotopy theory to algebraic K-theory of arbitrary qcqs schemes, and prove an algebraic analogue of Snaith theorem, which says that K-theory is obtained from the Picard stack by inverting the Bott element.

Orateur: Prof. Ryomei Iwasa (Université Paris-Saclay)

Vidéo

12:30 → 14:30 Lunch break

14:30 → 15:45 Efimov K-theory (3/3)

The scope of algebraic K-theory has been greatly extended by Alexander Efimov, who showed that it can be defined in a much broader categorical context than previously realized. This extended scope allows for precise connections with topology and analysis. I will give an introduction to Efimov's theory, and then focus on applications to complex geometry, the latter being joint work with Peter Scholze.

Orateur: Prof. Dustin Clausen (University of Copenhagen & IHES)

Vidéo

15:45 → 16:30 Coffee break / Q&A

16:30 → 17:30 Gromov-Witten Invariants in A1-homotopy Theory

The number of degree d rational plane curves through 3d-1 generally chosen points is independent of the generally chosen points over the complex numbers. (There is 1 line through 2 points, 1 conic through 5, 12 rational degree 3 curves through 8...) An early success of Gromov--Witten theory was to give a recursive formula for these invariants. Over the real numbers, there are invariants due to Jean-Yves Welschinger, Cheol-Hyun Cho and Jake Solomon giving an open Gromov-Witten invariant equal to a signed count of real curves. It is a feature of A1-homotopy theory that analogous real and complex results can indicate the presence of a common generalization, valid over a general field. We develop an A1-degree, following Fabien Morel, which in certain cases is the pushforward in Hermitian K-theory. We compute the A1-degree of an evaluation map on the Kontsevich moduli space of stable rational maps to obtain a count of genus 0 curves on certain del Pezzo surfaces through the appropriate number of marked points. This count is valid for any field k of characteristic not 2 or 3. In particular, we define and compute some Gromov--Witten invariants over a finite field. This is joint work with Jesse Kass, Marc Levine, and Jake Solomon. Time permitting, the talk will include joint work with Erwan Brugallé.

Orateur: Prof. Kirsten Wickelgren (Duke University)

Vidéo

jeudi 20 juillet

08:45 → 09:15 Welcome coffee

09:15 → 10:30 Arbeitsgemeinschaft 7

Definition of syntomic cohomology, basic properties and examples.

Orateur: Vova Sosnilo (University of Regensburg)

10:30 → 11:15 Coffee break / discussion

11:15 → 12:15 Prismatic Cohomology Relative Delta-Rings

Prismatic cohomology originally arose from the study of topological cyclic homology and comes in two flavors: Absolute prismatic cohomology, which is more tightly connected to topological cyclic homology, and the comparatively more computable relative prismatic cohomology, where one works relative to a fixed base prism. In this talk, I want to explain "prismatic cohomology relative delta-rings", a mutual generalization interpolating between the two.

Orateur: Prof. Achim Krause (University of Münster)

Vidéo

12:15 → 14:15 Lunch break (buffet at IHES)

14:15 → 15:30 Arbeitsgemeinschaft 8: Selmer K-theory and its motivic filtration

I will explain the construction of Selmer K-theory, its relation to étale K-theory, and the motivic filtration on its p-adic completion by syntomic cohomology

Orateur: Akhil Mathew (University of Chicago)

15:30 → 16:30 Coffee break / Q&A

16:30 → 17:30 Twisted K-theory in Motivic Homotopy Theory

In this talk, we will speak about algebraic K-theory of vector bundles twisted by a Brauer class, and its place in motivic homotopy theory. In particular, we will discuss a new approach to the motivic spectral sequence for twisted K-theory, constructed earlier by Bruno Kahn and Marc Levine. The talk is based on joint work with Elden Elmanto and Denis Nardin.

Orateur: Prof. Maria Yakerson (CNRS & Sorbonne Université)

Vidéo

vendredi 21 juillet

08:45 → 09:15 Welcome coffee

09:15 → 10:30 Motivic Stable Homotopy Theory (3/3)

In joint work with Toni Annala and Marc Hoyois, we have developed motivic stable homotopy in broader generality than the theory initiated by Voevodsky, so that non-$A^1$-invariant theories can also be captured. I’ll describe this, bearing in mind its connection to algebraic K-theory and p-adic cohomology such as syntomic cohomology. The course is divided roughly into three parts.
Foundations: The goal of this part is to grasp the notion of $P^1$-spectrum, which forms the basic framework of motivic stable homotopy theory.
Techniques: The goal of this part is to understand our main technique, P-homotopy invariance, which allows us to do a homotopy theory in algebraic geometry while keeping the affine line $A^1$ non-contractible.
Applications: In this part, we apply our motivic homotopy theory to algebraic K-theory of arbitrary qcqs schemes, and prove an algebraic analogue of Snaith theorem, which says that K-theory is obtained from the Picard stack by inverting the Bott element.

Orateur: Prof. Ryomei Iwasa (Université Paris-Saclay)

Vidéo

10:30 → 11:00 Coffee break / discussion

11:00 → 12:00 Presentation Lemmas in Mixed Characteristic

In his proof of the equal characteristic case of the Gersten conjecture for algebraic K-theory, Quillen used a geometric presentation lemma that may be viewed as a refinement of the Noether normalization theorem. Gabber, Gros--Suwa, and others have subsequently generalized Quillen's presentation lemma to establish Gersten-type phenomena for other cohomological functors. I will discuss presentation lemmas in mixed characteristics and present some of their consequences for the study of torsors under reductive groups.

Orateur: Prof. Kęstutis Česnavičius (CNRS & Université Paris-Saclay)

Vidéo