Methods of Noncommutative Geometry

elective seminars

Blended learning

The seminar aims to cover selected methods of homological algebra and functional analysis applied in the geometry and topology of manifolds, classical and quantum dynamical systems and representation theory. The main focus will be K-theory, (co)homological invariants, non-classical symmetries derived from classical geometry and different versions of the index theorem.

The topics of the seminar will include the following tools used in the study of topological spaces and operator algebras:

- spectral geometry of the Dirac operator,

- K-theory of topological spaces and C * -algebras,

- cyclic homology and the Chern character,

- K-homology and index pairing,

- quantum groups and Hopf-Galois theory.

- M. Khalkhali, Basic Noncommutative Geometry

- A. Connes, Noncommutative Geometry

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

- J. L. Loday, Cyclic Homology

- E. Abe, Hopf Algebras

- S. Neshveyev, L. Tuset. Compact Quantum Groups and Their Representation Categories

Knowledge and skills:

1. Understanding the classical relationships between spaces and algebras.

2. Knowledge of the basic concepts of noncommutative geometry.

3. Knowledge of the basic methods of algebra, homological algebra and functional analysis used in solving problems of topology, geometry, representation theory and the theory of operator algebras.

4. Ability to prepare and deliver lectures of varying degrees of difficulty on the basis of the assigned reading.

Social competence:

1. Ability to cooperate with representatives of the physical sciences in building mathematical models in physics (e.g., noncommutative versions the Standard Model of elementary particles).

2. Ability to deliver mathematical lectures understandable for representatives of other sciences and lectures for mathematicians on mathematical models in physics.

3. Ability to popularize modern mathematics.

Delivering a talk at the seminar.