NSCPHD1291 Groups, boundary actions and group C*-algebras

Volume 2014/2015
Content

The Poisson-­Furstenberg boundaries of a group are probabilistic objects,
connected with the study of random walks on the group. The action of a group on its Furstenberg and Poisson boundaries encodes a great deal of
information about geometric and analytic properties of the group (such as
amenability, growth and exactness), as well as operator algebraic properties of
the associated (reduced) group C*‐algebra.

In a spectacular very recent work, Breuillard, Kennedy, Kalantar and
Ozawa have proved simplicity of the reduced group C*-­algebra of a large
class of discrete groups using a new characterisation of this property in
terms of boundary actions. The study of C*-­simplicity and uniqueness of
trace property, two properties of groups grounded in C*-­algebra theory
has captured the interest of mathematicians for the last four decades.
Understanding these properties has intriguing deep connections with the
structure of the given group. For example, in a C*-­simple group and in a
group with the unique trace property, the only amenable normal subgroup
is the trivial one.

The purpose of this PhD Master class is to present the recent developments and applications of the theory of Poisson-­Furstenberg boundaries of a group, both from the probabilistic approach, a topic in which Kaimanovich is a world-­renowned expert, and from the new operator algebraic perspective, developed by Breuillard, Kennedy, Kalantar and Ozawa, that resulted in the settling of a number of longstanding open problems. Kaimanovich (currently holding a prestigious Canada Research Chair), Breuillard (highly acclaimed for his invited session lecture at the International Congress of Mathematicians, Seoul, 2014),
and Kennedy (a brilliant young mathematician, currently holding a canadian NSERC grant, who will be a long-term visitor of the department in the Spring of 2015) will each deliver mini-­series of lectures. Additional lectures on the structure of inner amenable groups will be delivered by Tucker-­Drob, another very young, yet well-­recognized mathematician. All speakers are confirmed.

These topics are at the forefront of current reasearch in geometric group
theory and operator algebras, and the Master class will benefit a broad
range of PhD students from our department, as well as PhD students
from other universities in Denmark and abroad.

Magdalena Musat is responsible for the overall organization of the Master
class, including the main lecture series, as well as problem/discussion
sessions. Notes from the lectures will be made available.

Learning Outcome

The learning outcome for the students

knowledge:

- fundamentals of the theory of boundaries of groups and random walks on groups.
- the basics and more advanced theory of group C*-algebras, 

skills: proof techniques in these topics

competencies: ability to read research papers on these topics

Lectures and problem/discussion sessions.
  • Category
  • Hours
  • Lectures
  • 25
  • Preparation
  • 30
  • Total
  • 55
Credit
2 ECTS
Type of assessment
Course participation
Marking scale
passed/not passed
Censorship form
No external censorship
Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.