Arbeitsgemeinschaft 1: Introduction

• Matthew Morrow (CNRS)

Room: Marilyn and James Simons Conference Center An introduction to the Arbeitsgemeinschaft on syntomic cohomology, and an overview of parts of the theory in characteristic p.

Localizing Invariants and Algebraic K-theory (1/3)

• Georg Tamme (University of Mainz)

Room: Marilyn and James Simons Conference Center 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.

Introduction to Prismatic Cohomology (1/4)

• Johannes Anschütz (University of Bonn)

Room: Marilyn and James Simons Conference Center 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

A Homotopical Approach to Crystalline Cohomology

• Moritz Kerz (University of Regensburg)

Room: Marilyn and James Simons Conference Center 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.)

Introduction to Prismatic Cohomology (2/4)

• Johannes Anschütz (University of Bonn)

Room: Marilyn and James Simons Conference Center 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

Localizing Invariants and Algebraic K-theory (2/3)

• Georg Tamme (University of Mainz)

Room: Marilyn and James Simons Conference Center 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.

[Videoconference] On the Motivic Cohomology of Schemes (1/3)

• Elden Elmanto (University of Toronto)

Room: Marilyn and James Simons Conference Center 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.

Structures and Computations in the Motivic Stable Homotopy Categories

• Hana Jia Kong (Institute for Advanced Study)

Room: Marilyn and James Simons Conference Center 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.

Introduction to Prismatic Cohomology (3/4)

• Johannes Anschütz (University of Bonn)

Room: Marilyn and James Simons Conference Center 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

Arbeitsgemeinschaft 2

• Noah Riggenbach (Northwestern University)

Room: Marilyn and James Simons Conference Center Definition of syntomic cohomology of p-complete (animated) rings, basic properties and examples.

Localizing Invariants and Algebraic K-theory (3/3)

• Georg Tamme (University of Mainz)

Room: Marilyn and James Simons Conference Center 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.

Algebraic K-Theory in Geometric Topology

• Thomas Nikolaus (University of Münster)

Room: Marilyn and James Simons Conference Center 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.

Introduction to Prismatic Cohomology (4/4)

• Johannes Anschütz (University of Bonn)

Room: Marilyn and James Simons Conference Center 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

Arbeitsgemeinschaft 3

• Emanuel Reinecke (MPIM Bonn)

Room: Marilyn and James Simons Conference Center Syntomic first Chern class; first part of proof that it is an equivalence after p-completion.

[Videoconference] On the Motivic Cohomology of Schemes (2/3)

• Elden Elmanto (University of Toronto)

Room: Marilyn and James Simons Conference Center 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.

Prismatic Cohomology of Commutative Ring Spectra

• Arpon Raksit (Massachusetts Institute of Technology)

Room: Marilyn and James Simons Conference Center 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.

Arbeitsgemeinschaft 4

• Shubhodip Mondal (MPIM Bonn)

Room: Marilyn and James Simons Conference Center Second step of proof that syntomic first Chern class is equivalence after p-completion.

[Videoconference] On the Motivic Cohomology of Schemes (3/3)

• Elden Elmanto (University of Toronto)

Room: Marilyn and James Simons Conference Center 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.

Efimov K-theory (1/3)

• Dustin Clausen (University of Copenhagen & IHES)

Room: Marilyn and James Simons Conference Center 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.

The Pro-cdh Topology

• Shane Kelly (University of Tokyo)

Room: Marilyn and James Simons Conference Center 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.

Arbeitsgemeinschaft 5

• Bogdan Zavyalov (IAS)

Room: Marilyn and James Simons Conference Center Third step of proof that syntomic first Chern class is equivalence after p-completion.

Motivic Stable Homotopy Theory (1/3)

• Ryomei Iwasa (Université Paris-Saclay)

Room: Marilyn and James Simons Conference Center 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.

Efimov K-theory (2/3)

• Dustin Clausen (University of Copenhagen & IHES)

Room: Marilyn and James Simons Conference Center 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.

K-theoretic Localization Theorem

• Charanya Ravi (Indian Statistical Institute)

Room: Marilyn and James Simons Conference Center 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.

Arbeitsgemeinschaft 6

• Guido Bosco (MPIM Bonn)

Room: Marilyn and James Simons Conference Center Construction of syntomic-to-étale comparison maps

Motivic Stable Homotopy Theory (2/3)

• Ryomei Iwasa (Université Paris-Saclay)

Room: Marilyn and James Simons Conference Center 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.

Efimov K-theory (3/3)

• Dustin Clausen (University of Copenhagen & IHES)

Room: Marilyn and James Simons Conference Center 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.

Gromov-Witten Invariants in A1-homotopy Theory

• Kirsten Wickelgren (Duke University)

Room: Marilyn and James Simons Conference Center 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.

Arbeitsgemeinschaft 7

• Vova Sosnilo (University of Regensburg)

Room: Marilyn and James Simons Conference Center Definition of syntomic cohomology, basic properties and examples.

Prismatic Cohomology Relative Delta-Rings

• Achim Krause (University of Münster)

Room: Marilyn and James Simons Conference Center 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.

Arbeitsgemeinschaft 8: Selmer K-theory and its motivic filtration

• Akhil Mathew (University of Chicago)

Room: Marilyn and James Simons Conference Center 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

Twisted K-theory in Motivic Homotopy Theory

• Maria Yakerson (CNRS & Sorbonne Université)

Room: Marilyn and James Simons Conference Center 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.

Motivic Stable Homotopy Theory (3/3)

• Ryomei Iwasa (Université Paris-Saclay)

Room: Marilyn and James Simons Conference Center 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.

Presentation Lemmas in Mixed Characteristic

• Kęstutis Česnavičius (CNRS & Université Paris-Saclay)

Room: Marilyn and James Simons Conference Center 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.