NSCPHD1291 Groups, boundary actions and group C*-algebras
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.
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
- Category
- Hours
- Lectures
- 25
- Preparation
- 30
- Total
- 55
Please register at: musat@math.ku.dk (or at the course homepage: http://www.math.ku.dk/english/research/conferences/2015/group_c_algebras)
- 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.
Course information
- Language
- English
- Course code
- NSCPHD1291
- Credit
- 2 ECTS
- Level
- Ph.D.
- Duration
- Placement
- Spring
- Schedule
- Week 16: 13-17 April 2015
- Continuing and further education
- Study board
- Natural Sciences PhD Committee
Contracting department
- Department of Mathematical Sciences
Course responsibles
- Magdalena Elena Musat (5-7f87857386527f73867a407d8740767d)
Lecturers
Vadim Kaimanovich, University of Ottawa
Emmanuel Breulliard, Univeriste Paris
Robin Tucker-Drob, Rutgers University
Matthew Kennedy, Carleton University