## Infinitesimal categorical Torelli: an application to Kuznetsov' Fano 3-folds conjecture

In this talk, we introduce the infinitesimal categorical Torelli for Fano 3-folds. We obtain a commutative diagram connecting the infinitesimal spaces

of the geometric spaces and the categories. These are joint works with Jacovskis Augustinas, Zhiyu Liu, and Shizhuo Zhang. In a joint work with

Shizhuo Zhang, we give a simple new disproof of Kuznetsov's Fano 3-folds conjecture for degree 2 and index two Fano 3-fold based on infinitesimal

categorical Torelli theorems.

## Semi-infinite t-structures via hyperbolic localization

For any nice enough scheme with an action of the multiplicative group it is possible to define a t-structure on a certain category of monodromic D-modules using hyperbolic localization. I will discuss the construction of these t-structures as well as an application to geometric representation theory. Specifically, I will explain how the semi-infinite IC sheaf associated to a semisimple group arises as an intermediate extension in this t-structure for Drinfeld's Zastava spaces.

## Cohomological Hall algebras and P=W conjecture

Let S be a smooth surface, and Coh(S) the moduli stack of properly supported coherent sheaves on S. One can equip the Borel-Moore homology of Coh(S) with a convolution algebra structure; this is called the cohomological Hall algebra (CoHA) of S. While understanding CoHA algebraically is more or less hopeless for a general S, I will give an explicit presentation of its part, which corresponds to sheaves with zero-dimensional support.

## The tail of the colored Jones polynomial for arborescent knots

## Microlocal Riemann-Hilbert correspondence

For complex manifolds, the Riemann-Hilbert correspondence generalizes the classical correspondence between finite dimensional local systems and D-modules which are coherent as O-modules to perverse sheaves and regular holonomic D-modules. These later objects are in fact microlocal in nature that they can be regarded as living on the cotangent bundles, and the correspondence admits a microlocalization as well.

## How algebraic is space?

A phenomena in topology is said to be stable if it occurs in all sufficiently high dimensions. As discovered by Quillen over five decades ago, such phenomena are closely related to number theory, and can often be described in terms of arithmetic objects known as formal groups. Unfortunately, in general this dictionary is not quite a one-to-one and many periodicities one sees on the arithmetic side become broken and more complex in the world of topology.

## Why can Kontsevich's invariants detect exotic phenomena?

In topology, the difference between the category of smooth manifolds and the category of topological manifolds has always been a delicate and intriguing problem, called the "exotic phenomena". The recent work of Watanabe (2018) uses the tool "Kontsevich's invariants" to show that the group of diffeomorphisms of the 4-dimensional ball, as a topological group, has non-trivial homotopy type. In contrast, the group of homeomorphisms of the 4-dimensional ball is contractible.

## Analytic geometry and p-adic automorphic forms

## A Morse-Bott approach to the equivariant homotopy type

Morse theory provides an effective way to calculate the homology of smooth manifolds, in terms of critical points of a function and its gradient flow.

## Infinite discrete groups: geometry, analysis and probability

## Analytic Stacks

The purpose of this course is to propose new foundations for analytic geometry. The topics covered are as follows:

1. Light condensed abelian groups.

2. Analytic rings.

3. Analytic stacks.

4. Examples.

Lectures will be given by Dustin Clausen at IHES and Peter Scholze at MPI, and broadcast live at the other location.

We also plan to make the lectures accessible by Zoom, and record them.

Mi, 10(c.t.) - 12 Uhr, und Fr, 10(c.t.) - 12 Uhr, MPI-Hörsaal

## Analytic Stacks

The purpose of this course is to propose new foundations for analytic geometry. The topics covered are as follows:

1. Light condensed abelian groups.

2. Analytic rings.

3. Analytic stacks.

4. Examples.

Lectures will be given by Dustin Clausen at IHES and Peter Scholze at MPI, and broadcast live at the other location.

We also plan to make the lectures accessible by Zoom, and record them.

Mi, 10(c.t.) - 12 Uhr, und Fr, 10(c.t.) - 12 Uhr, MPI-Hörsaal

## Analytic Stacks

The purpose of this course is to propose new foundations for analytic geometry. The topics covered are as follows:

1. Light condensed abelian groups.

2. Analytic rings.

3. Analytic stacks.

4. Examples.

Lectures will be given by Dustin Clausen at IHES and Peter Scholze at MPI, and broadcast live at the other location.

We also plan to make the lectures accessible by Zoom, and record them.

Mi, 10(c.t.) - 12 Uhr, und Fr, 10(c.t.) - 12 Uhr, MPI-Hörsaal

## Analytic Stacks

1. Light condensed abelian groups.

2. Analytic rings.

3. Analytic stacks.

4. Examples.

Lectures will be given by Dustin Clausen at IHES and Peter Scholze at MPI, and broadcast live at the other location.

We also plan to make the lectures accessible by Zoom, and record them.

Mi, 10(c.t.) - 12 Uhr, und Fr, 10(c.t.) - 12 Uhr, MPI-Hörsaal

## Analytic Stacks

1. Light condensed abelian groups.

2. Analytic rings.

3. Analytic stacks.

4. Examples.

Lectures will be given by Dustin Clausen at IHES and Peter Scholze at MPI, and broadcast live at the other location.

We also plan to make the lectures accessible by Zoom, and record them.

Mi, 10(c.t.) - 12 Uhr, und Fr, 10(c.t.) - 12 Uhr, MPI-Hörsaal

## Analytic Stacks

1. Light condensed abelian groups.

2. Analytic rings.

3. Analytic stacks.

4. Examples.

Lectures will be given by Dustin Clausen at IHES and Peter Scholze at MPI, and broadcast live at the other location.

We also plan to make the lectures accessible by Zoom, and record them.

Mi, 10(c.t.) - 12 Uhr, und Fr, 10(c.t.) - 12 Uhr, MPI-Hörsaal

## Analytic Stacks

1. Light condensed abelian groups.

2. Analytic rings.

3. Analytic stacks.

4. Examples.

Lectures will be given by Dustin Clausen at IHES and Peter Scholze at MPI, and broadcast live at the other location.

We also plan to make the lectures accessible by Zoom, and record them.

Mi, 10(c.t.) - 12 Uhr, und Fr, 10(c.t.) - 12 Uhr, MPI-Hörsaal

## Analytic Stacks

1. Light condensed abelian groups.

2. Analytic rings.

3. Analytic stacks.

4. Examples.

Lectures will be given by Dustin Clausen at IHES and Peter Scholze at MPI, and broadcast live at the other location.

We also plan to make the lectures accessible by Zoom, and record them.

Mi, 10(c.t.) - 12 Uhr, und Fr, 10(c.t.) - 12 Uhr, MPI-Hörsaal

## Selected Topics in Differential Geometry - The classical Plateau Problem

## Stably exotic surfaces and Khovanov homology

(Virtual)

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