Introduction to the Baum-Connes Conjecture 1000-1S24HBM

We present an introduction to Baum-Connes hypothesis describing a deep relationship between certain geometric and analytic objects associated to discrete groups (the relationship identifies K-theory of the group C*-algebra with the equivariant K-homology of the suitable classifying space for group actions).

In particular we will carefully introduce all the notions appearing in the formulation of the conjecture. The course will include the following topics:

Definition of a C*-algebra and C*-algebraic K-theory (definitions, examples, functorial properties, Bott periodicity).

C*-algebras of discrete groups: construction, examples, properties.

Classifying spaces for group actions (definitions and examples)) and equivariant K-homology..

(Equivariant) Kasparov’s KK-theory.

Definitions of the analytic assembly map and concrete examples.

Sketch of the state-of-art regarding the Baum-Connes conjecture, its alternative versions and generalizations (hypothesis for groupoids, hypothesis with coefficients).

Main fields of studies for MISMaP

mathematics

Type of course

elective seminars

Prerequisites (description)

We expect the knowledge contained in elementary courses in functional analysis and topology (in particular familiarity with the basics of the Hilbert space operators and fundamental topological notions). We do not assume any knowledge of operator algebras or advanced geometry, all relevant notions will be introduced and explained during the course.

Course coordinators

Piotr Nowak
Adam Skalski

Learning outcomes

Student knows how to define C*-algebras and C*-algebraic K-theory, knows the functorial properties of the latter and is able to apply them in concrete examples.

Can construct group C*-algebras, knows which group properties the relevant C*-algebra remembers.

Knows topological constructions related to classifying spaces for group actions and can present concrete examples.

Can define the analytic assembly map and formulate the `two sides’ of the Baum-Connes conjecture (surjectivity and injectivity of the assembly map). Is able to discuss the consequences of each side.

Understands the relationship between the Baum-Connes conjecture and the Atiyah-Singer index theory.

Knows the current state of knowledge regarding the conjecture.

Assessment criteria

Student presentations.

Active and regular participation in the meetings.

Bibliography

Alain Valette, Introduction to the Baum-Connes conjecture. From notes taken by Indira Chatterji. With an appendix by Guido Mislin. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2002. x+104 pp. (main source)

Maria Gomez Aparicio, Pierre Julg, Alain Valette, The Baum-Connes conjecture: an extended survey. in Advances in noncommutative geometry—on the occasion of Alain Connes' 70th birthday, 127–244, Springer, Cham, [2019], ©2019.

N. E. Wegge-Olsen, K-Theory and C*-Algebras: A Friendly Approach,

Guido Mislin and Alain Valette, Proper group actions and the Baum-Connes conjecture. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2003. viii+131 pp

Additional information

Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:

Master's degree, second cycle programme, Mathematics

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

Description of 1000-1S24HBM in USOSweb