Learning Seminar

This is the webpage of a Learning Seminar at Purdue on Groups, Geometry, Topology, and Dynamics for Fall 2024. We will post abstracts of talks and some lecture notes.

Organizers/Instructors: Lvzhou Chen, Ben McReynolds and Sam Nariman. 

Fall 2024

Regular Meeting Time: Fridays 10:30-11:20 am.

Location: SCHM 314

Module 1: Geometry, Lie groups, and Lattices

I (=Ben) will review some material from the lectures on Lie groups and lattices from the spring. I will discuss the geometric/algebraic dictionary between the algebraic structures defining a lattice and the geometric properties of the associated finite volume locally symmetric orbifold. I will focus on lattices in PSL(2,C) which give rise to complete, finite volume hyperbolic 3-orbifolds. The geometric structures that we will discuss will be geodesics and totally geodesic, immersed hyperbolic 2-orbifolds. I will discuss how one can generate all of the geodesics via the commensurator and special hyperbolic elements called absolutely primitive elements. I will end with a discussion of a conjectural characterization of arithmeticity based on previous work with Jean Lafont which was extended by my former student Nick Miller in his PhD thesis which connects with (almost) arithmetic progressions in the primitive geodesic length set.

Lecture 1 (Sept. 20)

Abstract: I will review the construction of arithmetic lattices in PSL(2,R) and PSL(2,C). I will discuss how to associate to Zariski dense subgroup of PSL(2,C) a pair of algebraic invariants called the invariant trace field and invariant quaternion algebra. These invariants will play a prominent role in the second lecture when we discuss geometric connections.

Lectures canceled on Sept. 27 and Oct. 4

Lecture 2 (Oct. 11)

Module 2: Patterson-Sullivan Theory

Patterson-Sullivan theory was developed by Patterson, extended by Sullivan, and greatly generalized by Benoist and Quint. This lecture series will focus on constant negative curvature and then on variable negative curvature. I will give a detailed overview of the topic starting from the requisite background material. I will end with a discussion of some applications, touching on work of Besson--Courtois--Gallot, Kapovich, and work with Farb--Connell and Connell--Wang. The last is a recent paper where we introduce a flow that serves as a continuous version of the natural or barycenter map defined by Besson--Courtois--Gallot.

Module 3: Representations of Surface Groups into Reductive Lie groups

This lecture series will focus on the representation theory of surfaces into reductive Lie groups. We will start with PSL(2,R) and PSL(2,C) which have an extensive history with connections to Teichmuller theory and hyperbolic geometry. We then will discuss the representation theory into higher dimensional Lie groups; this area is sometimes referred to as higher Teichmuller theory. I will survey (with some details) the work of Goldman, Fock-Goncharov, Bradlow--Garcia-Prada--Gothen, and Burger--Iozzi--Wienhard.