Instructor: Prof. Giulio Tiozzo
Email: tiozzo@math.utoronto.ca

Course Dates: September 12th - December 7th, 2023
Mid-Semester Break: October 9th - 13th, 2023
Lecture Times: Tuesdays 10:00 AM - 12:00 PM | Thursdays 10:00 AM - 11:00 AM (ET)
Office Hours: TBA

Registration Fee: PSU Students - Free | Other Students - CAD$500

Capacity Limit: 50 students

Format: Hybrid synchronous delivery

• In-Person: Fields Institute Room 309 (Stewart Library)
• Online: via Zoom

Course Description

This class will focus on properties of group actions from a probabilistic point of view, investigating the relations between the dynamics, measure theory and geometry of groups. The class is in preparation to the Winter 2024 Thematic Program on Randomness and Geometry at the Fields Institute.

We will start with a brief introduction to ergodic theory, discussing measurable transformations and the basic ergodic theorems. Then we will approach random walks on matrix groups and lattices in Lie groups, following the work of Furstenberg. Topics of discussion will be: positivity of drift and Lyapunov exponents. Stationary measures. Geodesic tracking. Entropy of random walks. The Poisson-Furstenberg boundary. Applications to rigidity. We will then turn to a similar study of group actions which do not arise from homogeneous spaces, but which display some features of negatively curved spaces: for instance, hyperbolic groups (in the sense of Gromov) and groups acting on hyperbolic spaces. This will lead us to applications to geometric topology: in particular, to the study of mapping class groups and Out(F_N).

Course expectations: It is expected that students attend all lectures and complete the homework assignments (one every 2 weeks). Auditing may be allowed upon authorization from the instructor.

Prerequisites: An introduction to measure theory and/or probability, basic topology and basic group theory. No previous knowledge of geometric group theory or Teichmueller theory is needed.

Course Materials: Lecture notes and other materials will be posted on the course website. Students are responsible for checking the course website on a regular basis for news and updates.

Evaluation method:

• (35 %) Homework
• (30 %) Final presentation
• (35 %) Final written project

You can find all lecture recordings here: https://www.youtube.com/playlist?list=PLArBKNfJxuukpVdd17XhdvreKKgZVg5Zy