Wenhan Dai

Recent Advances in Algebraic K-theory

IHES 2023 Summer School

• Time: July 10 to 21, 2023.
• Organizers: 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 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.

Lecture Notes (in chronological order)

I focus on topics on prismatic cohomology, hence the following is a selected schedule only. Please use with caution and do not disseminate.

July 11

• Johannes Anschütz (Univ. of Bonn) – Introduction to Prismatic Cohomology (1/4). (video, notes)
  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.

July 11

• Johannes Anschütz (Univ. of Bonn) – Introduction to Prismatic Cohomology (2/4). (video, notes)
  This is a follow-up of the previous session.

July 12

• Johannes Anschütz (Univ. of Bonn) – Introduction to Prismatic Cohomology (3/4). (video, notes)
  This is a follow-up of the previous session.

July 13

• Johannes Anschütz (Univ. of Bonn) – Introduction to Prismatic Cohomology (4/4). (video, notes)
  This is a follow-up of the previous session.