Home /  Geometry and Dynamics in Higher Rank Lie Groups (UC Berkeley)

Summer Graduate School

Geometry and Dynamics in Higher Rank Lie Groups (UC Berkeley) July 14, 2025 - July 25, 2025
Parent Program:
Location: Lewis Hall, UC Berkeley
Organizers Richard Canary (University of Michigan), Sara Maloni (University of Virginia), Wenyu Pan (University of Toronto; University of Toronto), Cagri Sert (University of Warwick), LEAD Tengren Zhang (National University of Singapore)
Lecturer(s)

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Teaching Assistants(s)

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Speaker(s)

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Description
Image
<p>Flats and hyperbolic planes in a higher rank symmetric space</p> Drawn by Steve Trettel.

Lie groups are central objects in modern mathematics; they arise as the automorphism groups of many homogeneous spaces, such as flag manifolds and Riemannian symmetric spaces. Often, one can construct manifolds locally modelled on these homogeneous spaces by taking quotients of their subsets by discrete subgroups of their automorphism groups. Studying such discrete subgroups of Lie groups is an active and growing area of mathematical research. The objective of this summer school is to introduce young researchers to a class of discrete subgroups of Lie groups, called Anosov subgroups. These subgroups are central objects in the study of higher Teichmuller theory, convex projective geometry, and character varieties. In this summer school, there will be an emphasis on discussing some of the dynamical tools that have recently been successfully used to study Anosov subgroups. The required background in dynamics, hyperbolic geometry, and Lie theory will also be discussed. Aside from providing a stimulating academic environment for learning about Anosov subgroups, this summer school also aims to be a relaxed and friendly space for participants to interact with fellow aspiring mathematicians. There will also be many opportunities for participants to consult with the expert lecturers of the summer school. Join us for an unforgettable fortnight of discovery and learning!

School Structure

There will be two lectures each day, as well as two problem sessions. 

Prerequisites

The following are the suggested prerequisites for the summer school.

(1) Algebra prerequisites

  • Groups and group actions: Part I of Abstract Algebra by David S. Dummit and Richard M. Foote
  • Linear algebra: Part III of Abstract Algebra by David S. Dummit and Richard M. Foote

(2) Topology prerequisites 

  • Point set topology: Chapters 1–7 of Topology by James Munkres
  • Covering spaces: Chapter 1 of Algebraic Topology by Allen Hatcher; Chapters 9–12 of Topology by James Munkres
  • Differential topology and Riemannian geometry: Chapters 1–14 of Introduction to Smooth Manifolds by John M. Lee (errata)

(3) Lie groups prerequisites

(4) Analysis prerequisites

  • Measure theory: Chapters 11–15 of Real Analysis by H.L. Royden

Application Procedure

For eligibility and how to apply, see the Summer Graduate Schools homepage.

Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • Lie groups

  • discrete groups

  • geometric structures

  • Teichmuller theory

  • entropy

  • ergodicity

  • mixing

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
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Schedule, Notes/Handouts & Videos
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Jul 14, 2025
Monday
08:40 AM - 09:00 AM
  Welcome to SLMath
09:00 AM - 10:15 AM
  Real hyperbolic spaces and their isometry groups
Pierre-Louis Blayac (Université de Strasbourg)
10:15 AM - 10:45 AM
  Morning Break
10:45 AM - 12:00 PM
  Real hyperbolic spaces and their isometry groups
Pierre-Louis Blayac (Université de Strasbourg)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
Aleksander Skenderi (University of Wisconsin-Madison)
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Suxuan Chen (Ohio State University)
Jul 15, 2025
Tuesday
09:00 AM - 10:15 AM
  Gromov hyperbolicity and convergence group actions
Pierre-Louis Blayac (Université de Strasbourg)
10:15 AM - 10:45 AM
  Morning Break
10:45 AM - 12:00 PM
  Gromov hyperbolicity and convergence group actions
Pierre-Louis Blayac (Université de Strasbourg)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
Aleksander Skenderi (University of Wisconsin-Madison)
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Suxuan Chen (Ohio State University)
Jul 16, 2025
Wednesday
09:00 AM - 10:15 AM
  Flag manifolds of SL(d,R) and transverse, Anosov, and relative Anosov groups
Pierre-Louis Blayac (Université de Strasbourg)
10:15 AM - 10:45 AM
  Morning Break
10:45 AM - 12:00 PM
  Flag manifolds of SL(d,R) and transverse, Anosov, and relative Anosov groups
Pierre-Louis Blayac (Université de Strasbourg)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
Aleksander Skenderi (University of Wisconsin-Madison)
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Suxuan Chen (Ohio State University)
Jul 17, 2025
Thursday
09:00 AM - 10:15 AM
  Symmetric space, Busemann functions, Gromov product, flows
Pierre-Louis Blayac (Université de Strasbourg)
10:15 AM - 10:45 AM
  Morning Break
10:45 AM - 12:00 PM
  Symmetric space, Busemann functions, Gromov product, flows
Pierre-Louis Blayac (Université de Strasbourg)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
Aleksander Skenderi (University of Wisconsin-Madison)
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Suxuan Chen (Ohio State University)
Jul 18, 2025
Friday
09:00 AM - 10:15 AM
  Structure of limit sets
Andrew Zimmer (University of Wisconsin-Madison)
10:15 AM - 10:45 AM
  Morning Break
10:45 AM - 12:00 PM
  Patterson-Sullivan measures in rank one
Andrew Zimmer (University of Wisconsin-Madison)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
Aleksander Skenderi (University of Wisconsin-Madison)
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Suxuan Chen (Ohio State University)
Jul 21, 2025
Monday
09:00 AM - 10:15 AM
  Patterson-Sullivan measures in higher rank
Andrew Zimmer (University of Wisconsin-Madison)
10:15 AM - 10:45 AM
  Morning Break
10:45 AM - 12:00 PM
  Affinity exponents and Hausdorff dimension of limit sets
Wenyu Pan (University of Toronto; University of Toronto)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
Aleksander Skenderi (University of Wisconsin-Madison)
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Suxuan Chen (Ohio State University)
Jul 22, 2025
Tuesday
09:00 AM - 10:15 AM
  Patterson-Sullivan measures in higher rank
Andrew Zimmer (University of Wisconsin-Madison)
10:15 AM - 10:45 AM
  Morning Break
10:45 AM - 12:00 PM
  Affinity exponents and Hausdorff dimension of limit sets
Wenyu Pan (University of Toronto; University of Toronto)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
Aleksander Skenderi (University of Wisconsin-Madison)
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Suxuan Chen (Ohio State University)
Jul 23, 2025
Wednesday
09:00 AM - 10:15 AM
  Patterson-Sullivan measures in higher rank
Andrew Zimmer (University of Wisconsin-Madison)
10:15 AM - 10:45 AM
  Morning Break
10:45 AM - 12:00 PM
  Stationary measures and Lyapunov dimension
Wenyu Pan (University of Toronto; University of Toronto)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Problem Session
Aleksander Skenderi (University of Wisconsin-Madison)
03:00 PM - 05:00 PM
  Visit to SLMath with Afternoon Tea and Group Photo
Jul 24, 2025
Thursday
09:00 AM - 10:15 AM
  Entropy rigidity for Hitchin representations
Andrew Zimmer (University of Wisconsin-Madison)
10:15 AM - 10:45 AM
  Morning Break
10:45 AM - 12:00 PM
  Stationary measures and Lyapunov dimension
Wenyu Pan (University of Toronto; University of Toronto)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
Aleksander Skenderi (University of Wisconsin-Madison)
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Suxuan Chen (Ohio State University)
Jul 25, 2025
Friday
09:00 AM - 10:15 AM
  Entropy growth argument for Bernoulli convolution
Wenyu Pan (University of Toronto; University of Toronto)
10:15 AM - 10:45 AM
  Morning Break
10:45 AM - 12:00 PM
  Entropy growth argument for Bernoulli convolution
Wenyu Pan (University of Toronto; University of Toronto)
12:00 PM - 01:30 PM
  Lunch
01:30 PM - 02:00 PM
  Participant Survey
02:00 PM - 03:30 PM
  Problem Session
Suxuan Chen (Ohio State University)
03:30 PM - 04:00 PM
  Afternoon Tea