Ulrich Bunke

Mathematische
Fakultät

__Universität
Regensburg__

93040
Regensburg,

Deutschland

mail:
ulrich.bunke at mathematik.uni-regensburg.de

Tel:
0941 943 2780

Sekretariat: B.
Lindner Tel: 0941
943 2757

**Lehrveranstaltungen im Wintersemester 2020/21**

AG-Seminar Do 12-14 Online

Description: The aim of this online seminar is to present and discuss recent results in the range from homotopy theory, K-theory to global analysis. Most of the talks will be given by guests.
Oberseminar Globale Analysis Mi 10-12 MA102

Introduction to coarse geometry Di, Fr 10:15-11:45 (online, via Zoom, start at Nov. 3)

Description: This course is an introduction to the geometry and homotopy theory of bornological coarse spaces. The course starts with the basic notions coarse and bornological structures
and introduces the category of bornological coarse spaces. The homotopy theory of bornological coarse spaces is then investigated through the notion of a coarse homology theory. The basic example for this course is the coarse ordinary homology but we also give a survey on versions of coarse K-homology. The course will study in detail how the homotopy theory of bornological coarse spaces is related with the homotopy theory of topological spaces via cone constructions and coarsifying spaces. As an outlook we will explain applications to global analysis, geometric group theory and other fields.

The reference for this course is the book ``Homotopy Theory with Bornological Coarse Spaces '' (Bunke/Engel), Lecture Notes in Math. 2269, but we will not use the language of infinity categories. Prerequisites are elements of categorical language and basics from general topology, and homological algebra, e.g. a one-semester introduction to algebraic topology.
Parallel to this course there will be a skript whose up to date version can be downloaded here (the link will be activated later).

Exercises: There will be additional weekly 2h-tutorial classes (organized by C. Winges) whose coordinates will be fixed later together with the participants.

Exam: We offer an 30 min. oral exam (online, if necessary)

C*-Algebras and Categories Mo 12:15-13:45 ** The
plan is to shift the time to Mo 16:15-17:45 ** (online, via Zoom, start at Nov. 2)

Description: This course provides an introduction to C*-algebras and C*-categories.
Its main goal is to understand the categories of these objects, e.g., how to form limits or colimits. The first half of the course consists of a self-contained introduction to C*-algebras
as a full subcategory of *-algebras based on the unicity of the norm on a C*-algebra.
We then discuss Gelfand duality for commutative C*-algebras and the continuous function calculus.
The second half of the course is devoted to C*-categories. As a basic example we will discuss the categories of Hilbert modules over C*-algebras and adjointable operators.
We will then explain adjunctions relating C*-algebras and categories or unital with non-unital versions.
Using universal constructions we will provide various examples of interesting C*-algebras and categories.
As a further application we will e.g. understand that forming crossed products of groups with C*-algebras can naturally be viewed as an operation taking of homotopy orbits in the category of C*-categories. As an outlook we will discuss further applications to coarse geometry and global analysis.

Prerequisites: We assume basic facts from the language of categories and functional analysis.

Skript:
Parallel to this course there will be a script whose up to date version can be downloaded here (the link will be activated later).

Exam: We offer an 30 min. oral exam (online, if necessary)

**Links:**

SFB Higher
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