The Atiyah-Singer index theorem via the heat kernel approach

Sprecher/Organisator: Prof. Dr. Johannes Ebert

Termin: Fr 10-12, SRZ 214

The aim of this seminar (aimed at PhD students in topology, differential geometry or functional analysis) is to understand the Atiyah--Singer index theorem, by the classical heat kernel proof and following the little book ''Elliptic operators, topology and asymptotic methods'' by John Roe. In the (unlikely) event that there is time and energy left, we proceed to the Atiyah--Patodi--Singer index theorem on manifolds with boundary.

Deligne Lusztig theory

Sprecher/Organisator: Prof. Dr. Eva Viehmann

Termin: Fr 10-12, SR1C

Deligne-Lusztig theory is an elegant geometric way to construct representations of finite groups of Lie type (i.e. the F_q-points of reductive groups over a finite field F_q). More precisely, we will study the l-adic cohomology of suitable subvarieties of flag varieties (so-called Deligne-Lusztig varieties) and their coverings. In particular one can construct in this way all representations of all finite simple groups of Lie type.

CRC mini course: Dynamics, comparison, and operator algebras

Sprecher/Organisatoren: JProf. Dr. Shirly Geffen and Prof. Dr. David Kerr

Termin: June 24 & July 1, 14:15-16:00 (two 45-minute talks on each of the two days), SRZ 216/217

The first Monday (June 24) will be a colloquium-style presentation in which we will explain the basic regularization phenomena in topological dynamics associated to structure of an additive (comparison, Rokhlin towers, density conditions) or multiplicative (entropy, mean dimension) nature and how these relate both technically and via analogy to properties of C*-algebras. On the second Monday (July 1) we will delve into the strategies for tackling some of the more subtle and vexing issues that arise in these connections between dynamics and C*-algebras.

Sheaves on Manifolds

Prof. Dr. Thomas Nikolaus

Mo/Do 10-12, SR 5

The lecture course addresses all investigators in the CRC whose projects are concerned with K-Theory.

The goal of the lecture is to explain the recent notion of continuous K-Theory and its relation to geometric topology. We will also cover Verdier duality, microsupport theory and six-functor formalisms for the categories of sheaves of spectra/chain complexes on topological spaces. Key is the recent formalism of locally rigid and dualizable stable infinity-categories (a la Gaitsgory-Rozenblyum, Lurie, Efimov, Clausen, ....). Part of the course will be an introduction to these abstract concepts. Also the shape of a  topos will be relevant, which can also serve as the basis of étale  homotopy theory (on demand we can cover basics of that as well). We plan to write lecture notes that will be made available here.

We will assume some familiarity with infinity-categories, but try to avoid overly technical details.