Random walks on groups and boundary theory 2026/2027
Random walks on groups and boundary theory 2026/2027
Autumn semester 2026/2027: 14-18 on Tuesdays, room P102
Office hours: Wednesdays 10-12 at IMPAN (office 424) or make an appointment by e-mail
Resources:
Harmonic functions and random walks on groups (by Ariel Yadin)
Summary of lecture:
Lecture 1: Markov chains: transient and recurrent random walks, harmonic functions and connection to martingales, Doob's optional stopping theorem;
Lecture 2: group constructions: subgroups, quotients, products, semidirect products, extensions, direct limits; nilpotent groups, solvable groups, amenable groups;
Lecture 3: group actions on topological and measure spaces; positive definite functions; invariant and tail sigma-algebras of the random walk, harmonic and parabolic functions, Poisson formula;
Problems:
Spring semester 2024/2025: 14-18 on Mondays, room 3220
Office hours: Wednesdays 10-12 at IMPAN (office 424) or make an appointment by e-mail
Resources:
Introduction to II1 factors (by Anantharaman and Popa)
Functional analysis notes (by Vaes and Wasilewski, spectral theorem is Lecture 4)
Notes on C*-algebras (by Putnam)
Problems:
Topics for presentations:
AF C*-algebras and Bratteli diagrams.
Graph C*-algebras and Cuntz-Krieger algebras.
Construction of type III von Neumann algebras.
Noncommutative L^p-spaces.
Completely positive maps and Stinespring's theorem.
Modules and von Neumann dimension.
Nuclearity of C*-algebras.
Fullness of von Neumann algebras.
Voiculescu's free Gaussian von Neumann algebras.
Haagerup's approximation property.
Hilbert modules over C*-algebras.
K-theory of C*-algebras.
Simplicity of group C*-algebras and Powers' averaging.
W roku akademickim 2023/2024 prowadziłem kurs Grafy kwantowe na Uniwersytecie Warszawskim.
Konsultacje: środa 10-12, pokój 424 IM PAN
In 2015 I was responsible for the tutorial classes for the course Functional Analysis II at the University of Warsaw. Here you can find all the problems with solutions (in Polish).