Title: Covering a curve
Abstract: Aimed at a general audience, the talk presents results and approaches to an unsolved problem in extremal geometry. 

Preceded and followed by musical contributions from the following artists:

Lory Aintablian (piano)

Remy van Dobben de Bruyn (vocals)

Dr. Gotthelf (piano)

Eva-Maria Hekkelman (cello)

Annika Kraasch-Tarnowsky (flute)

Davide Macera (piano)

Alexandre Maksoud (piano)

Sun Woo Park (piano)

David Prinz (piano)

Wyatt Reeves (piano)

Lola Thompson (vocals)

There will be an opportunity to enter MPIM from 17:00 until 17:30.

Description: https://people.mpim-bonn.mpg.de/scholze/ss2025_habiro.pdf

I will give an introduction to localizing invariants of dualizable categories. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then we will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum, and sketch some applications of this result.

I will give an introduction to localizing invariants of dualizable categories. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then we will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum, and sketch some applications of this result.

I will give an introduction to localizing invariants of dualizable categories. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then we will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum, and sketch some applications of this result.

I will give an introduction to localizing invariants of dualizable categories. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then we will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum, and sketch some applications of this result.

In this series of lectures, I will give an introduction to some fundamental concepts that will be relevant for this workshop:
localizing invariants of stable $\infty$-categories, Waldhausen $K$-theory, noncommutative motives, and presentable and dualisable categories

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

First Lecture: October 18, 2023

Recordings can be found on Youtube.