Cancelled Groups, Operator Algebras and Dynamics (GOADyn)

Course content

The course provides a thorough introduction to areas of very active current research concerning the interplay between groups, operator algebras and dynamics. Actions of groups on spaces give rise to C*-algebras via crossed products, thus being an important source of examples.

We study analytic and geometric properties of groups (e.g., amenability and exactness), reflecting into properties of associated C*-(and von Neumann) algebras (nuclearity, resp. exactness). Furthermore, we discuss a certain rigidity property of group actions on Hilbert spaces, Kazhdan's property (T). This gives rise to explicit constructions of expander graphs, that play an important role in theoretical computer science and pure mathematics. We present ideas of large scale (coarse) geometry, allowing us to conclude that expanders are incompatible with Euclidean geometry. Throughout the course, special emphasis is placed on examples of groups (e.g., free groups and discrete subgroups of Lie groups, such as SL(n, Z)) and their properties, as well as group-theoretic constructions (e.g., semi-direct products), leading to further interesting examples. These, in turn, lead to interesting examples of C*-algebras, as well as applications.

Education

MSc Programme in Mathematics

Learning outcome

After completing the course, the students will have:

Knowledge of the material mentioned in the description of the content.

Skills to to read and understand research papers concerning topics discussed in lectures.

The following competences

  • Have a good working knowledge of concrete examples of groups (such as free groups and discrete subgroups of Lie groups, e.g., SL(n, Z)), in terms of their analytic and geometric properties, and permanence properties.
  • Have a good understanding of how properties of groups reflect into properties of their associated group C*-(and von Neumann) algebras, as well as into properties of their actions on topological spaces.
  • Understand interesting examples of C*-algebras arizing as crossed products via actions of groups on spaces.
  • Have an overview of the study of groups viewed as geometric spaces, through the study of their Cayley graphs, using ideas of coarse geometry.
  • Understand applications of operator algebras to other areas of mathematics and theoretical computer science (e.g., expander graphs).
  • Master (at a satisfactory level) the fundamental results covered in the lectures, to the extent of understanding their proofs and be able to interconnect various results.

4 hours lectures, 2 hours exercises/discussion per week for 8 weeks.

Introduction to operator algebras.

Academic qualifications equivalent to a BSc degree is recommended.

Continuous feedback during the course of the semester
ECTS
7,5 ECTS
Type of assessment
Continuous assessment
Continuous assessment
Each student will give a presentation (up to 2 x 45 min) of material not covered in lectures, relevant to the topic of the course, coming either from a research paper or from a textbook.
Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
No external censorship
One internal examiner
Criteria for exam assessment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 32
  • Preparation
  • 138
  • Theory exercises
  • 16
  • Exam
  • 20
  • English
  • 206