Name: 2023 IHES Summer School - Recent Advances in Algebraic K-theory
Start: 2023-07-10T08:30:00+02:00
End: 2023-07-21T19:00:00+02:00
Location: IHES

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

de lundi 10 juillet 2023 (08:30) à vendredi 21 juillet 2023 (19:00)

lundi 10 juillet 2023

09:00 Welcome coffee and registration
Welcome coffee and registration
09:00 - 09:30
Room: Marilyn and James Simons Conference Center
09:30 Arbeitsgemeinschaft 1: Introduction - Matthew Morrow (CNRS)
Arbeitsgemeinschaft 1: Introduction
- Matthew Morrow (CNRS)
09:30 - 10:45
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.
10:45 Coffee break / discussion
Coffee break / discussion
10:45 - 11:15
Room: Marilyn and James Simons Conference Center
11:15 Localizing Invariants and Algebraic K-theory (1/3) - Georg Tamme (University of Mainz)
Localizing Invariants and Algebraic K-theory (1/3)
- Georg Tamme (University of Mainz)
11:15 - 12:30
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.
12:30 Lunch break (buffet at IHES)
Lunch break (buffet at IHES)
12:30 - 14:30
Room: Marilyn and James Simons Conference Center
14:30 Introduction to Prismatic Cohomology (1/4) - Johannes Anschütz (University of Bonn)
Introduction to Prismatic Cohomology (1/4)
- Johannes Anschütz (University of Bonn)
14:30 - 16:00
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
16:00 Coffee break / discussion
Coffee break / discussion
16:00 - 16:30
Room: Marilyn and James Simons Conference Center
16:30 A Homotopical Approach to Crystalline Cohomology - Moritz Kerz (University of Regensburg)
A Homotopical Approach to Crystalline Cohomology
- Moritz Kerz (University of Regensburg)
16:30 - 17:30
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.)
mardi 11 juillet 2023

08:45 Welcome coffee
Welcome coffee
08:45 - 09:15
Room: Marilyn and James Simons Conference Center
09:15 Introduction to Prismatic Cohomology (2/4) - Johannes Anschütz (University of Bonn)
Introduction to Prismatic Cohomology (2/4)
- Johannes Anschütz (University of Bonn)
09:15 - 10:45
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
10:45 Coffee break / discussion
Coffee break / discussion
10:45 - 11:15
Room: Marilyn and James Simons Conference Center
11:15 Localizing Invariants and Algebraic K-theory (2/3) - Georg Tamme (University of Mainz)
Localizing Invariants and Algebraic K-theory (2/3)
- Georg Tamme (University of Mainz)
11:15 - 12:30
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.
12:30 Lunch break
Lunch break
12:30 - 14:30
Room: Marilyn and James Simons Conference Center
14:30 [Videoconference] On the Motivic Cohomology of Schemes (1/3) - Elden Elmanto (University of Toronto)
[Videoconference] On the Motivic Cohomology of Schemes (1/3)
- Elden Elmanto (University of Toronto)
14:30 - 15:45
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.
15:45 Coffee break / discussion
Coffee break / discussion
15:45 - 16:30
Room: Marilyn and James Simons Conference Center
16:30 Structures and Computations in the Motivic Stable Homotopy Categories - Hana Jia Kong (Institute for Advanced Study)
Structures and Computations in the Motivic Stable Homotopy Categories
- Hana Jia Kong (Institute for Advanced Study)
16:30 - 17:30
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.
mercredi 12 juillet 2023

08:45 Welcome coffee
Welcome coffee
08:45 - 09:15
Room: Marilyn and James Simons Conference Center
09:15 Introduction to Prismatic Cohomology (3/4) - Johannes Anschütz (University of Bonn)
Introduction to Prismatic Cohomology (3/4)
- Johannes Anschütz (University of Bonn)
09:15 - 10:45
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
10:45 Coffee break / discussion
Coffee break / discussion
10:45 - 11:15
Room: Marilyn and James Simons Conference Center
11:15 Arbeitsgemeinschaft 2 - Noah Riggenbach (Northwestern University)
Arbeitsgemeinschaft 2
- Noah Riggenbach (Northwestern University)
11:15 - 12:30
Room: Marilyn and James Simons Conference Center Definition of syntomic cohomology of p-complete (animated) rings, basic properties and examples.
12:30 Lunch break
Lunch break
12:30 - 14:30
Room: Marilyn and James Simons Conference Center
14:30 Localizing Invariants and Algebraic K-theory (3/3) - Georg Tamme (University of Mainz)
Localizing Invariants and Algebraic K-theory (3/3)
- Georg Tamme (University of Mainz)
14:30 - 15:45
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.
15:45 Coffee break / discussion
Coffee break / discussion
15:45 - 16:30
Room: Marilyn and James Simons Conference Center
16:30 Algebraic K-Theory in Geometric Topology - Thomas Nikolaus (University of Münster)
Algebraic K-Theory in Geometric Topology
- Thomas Nikolaus (University of Münster)
16:30 - 17:30
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.
jeudi 13 juillet 2023

08:45 Welcome coffee
Welcome coffee
08:45 - 09:15
Room: Marilyn and James Simons Conference Center
09:15 Introduction to Prismatic Cohomology (4/4) - Johannes Anschütz (University of Bonn)
Introduction to Prismatic Cohomology (4/4)
- Johannes Anschütz (University of Bonn)
09:15 - 10:45
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
10:45 Coffee break / discussion
Coffee break / discussion
10:45 - 11:15
Room: Marilyn and James Simons Conference Center
11:15 Arbeitsgemeinschaft 3 - Emanuel Reinecke (MPIM Bonn)
Arbeitsgemeinschaft 3
- Emanuel Reinecke (MPIM Bonn)
11:15 - 12:30
Room: Marilyn and James Simons Conference Center Syntomic first Chern class; first part of proof that it is an equivalence after p-completion.
12:30 Lunch break
Lunch break
12:30 - 14:30
Room: Marilyn and James Simons Conference Center
14:30 [Videoconference] On the Motivic Cohomology of Schemes (2/3) - Elden Elmanto (University of Toronto)
[Videoconference] On the Motivic Cohomology of Schemes (2/3)
- Elden Elmanto (University of Toronto)
14:30 - 15:45
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.
15:45 Coffee break / discussion
Coffee break / discussion
15:45 - 16:30
Room: Marilyn and James Simons Conference Center
16:30 Prismatic Cohomology of Commutative Ring Spectra - Arpon Raksit (Massachusetts Institute of Technology)
Prismatic Cohomology of Commutative Ring Spectra
- Arpon Raksit (Massachusetts Institute of Technology)
16:30 - 17:30
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.
18:00 Buffet dinner at IHES
Buffet dinner at IHES
18:00 - 19:00
Room: Marilyn and James Simons Conference Center
vendredi 14 juillet 2023

09:00 July 14 - Public Holiday (French National Day)
July 14 - Public Holiday (French National Day)
09:00 - 17:00
Room: Marilyn and James Simons Conference Center
samedi 15 juillet 2023
dimanche 16 juillet 2023