Geometry and Topology Student Seminar

The Geometry & Topology Student Seminar aims to provide a relaxed environment where students can practice giving talks and learn about geometry and topology from other grad students and postdocs. We meet on Thursdays's 2-3PM in LCB 222.  Below is our schedule for Fall semester.  

Organizers: Thomas Hill, George Shaji.  

Fall 2025

Sept 18: George Shaji 

Title: A broad overview of infinite-type mapping class groups

Abstract: We will first look at the origin of the recent surge in interest in infinite type surfaces and then go over some of the basics of big mapping class groups using Rosendal's framework for Polish groups. As we explore the topic, we will also establish links to various other fields of research such as geometric and combinatorial group theory, dynamics and Teichmuller theory.

Sept 25: Harry Chen

Title: From fibrations of three manifolds to end periodic homeomorphisms of infinite type surfaces

Abstract: We begin with Thurston’s approach to representing cycles in H_2(M) with embedded surfaces and explore fibrations M^3 -> S^1 through the Thurston norm. We introduce the oriented sum of surfaces, and then illustrate how the sum of classes of fibrations within a fibered face again corresponds to a fibration. Finally, we show that limits of such sums give rise to homeomorphisms on infinite-type surfaces.

Oct 2: Minhua Cheng

Title: Dynamical cocycles and Lyapunov exponents

Abstract: Dynamical cocycles are a key tool for studying the linear behavior of a nonlinear system along its orbits. The Multiplicative Ergodic Theorem shows that this evolution has a very regular long-term structure: the space splits into directions where vectors grow or shrink at fixed exponential rates, called Lyapunov exponents. This structure helps us understand the stability of orbits and the geometry of the system. We will first introduce the key concepts with a motivating example and then explain the Multiplicative Ergodic Theorem. If time permits, we will ​explore​ some applications of this framework.

Oct 9: No seminar (Fall Break) 

Oct 16: Yibo Zhai

Title: Symbolic systems and counting geodesics

Abstract: Counting geodesics is one of the major problems in dynamical systems and geometry. There are many ways to count the number of geodesics, such as using Selberg's trace formula and Mirzakhani's method. We will discuss how to use the symbolic coding method to count geodesics of a given length. One can use the symbolic coding method to count geodesics on general surfaces, including some infinite genus surfaces. In this talk, we will show how to transfer the geodesic flow system to a symbolic system and then introduce the main idea of the counting method.

Oct 23: Sage Yeager

Title: Weyl Chamber Flows

Abstract: A diffeomorphism of a compact manifold M induces a Z-action on M by iteration, while a flow may be viewed as an R-action. In dynamics, such actions are often studied when they exhibit hyperbolic behavior. A natural next step is to consider higher-rank actions - those of Z^k or R^k - and to ask what hyperbolicity means in this broader setting. The Weyl chamber flow on a compact quotient of a semisimple Lie group G provides a fundamental example of a higher-rank hyperbolic action. In this talk, we will focus primarily on the case G = SL(3,R), before discussing how this example generalizes and briefly surveying related rigidity phenomena. 

Oct 30: Jeremy West (University of Oklahoma) 

Title: Σ-invariants: An overview

Abstract: For any group G, one can define the character sphere Σ(G) to be the vector space of homomorphisms from G to the real numbers, modulo positive scaling. The Σ-invariants of G are a filtration of Σ(G), defined over a series of papers by Bieri, Neumann, Strebel, and Renz to investigate when various finiteness properties of a group preserve to coabelian subgroups. In this talk, I will lay out initial definitions and the basic results, as well as showing various examples of how they can be used. 

Nov 6: Nilendu Das 

Title: Extremal length and Bers’ Inequality

Abstract: A quasi-Fuchsian hyperbolic 3-manifold, M is a very special class of infinite-volume hyperbolic 3-manifolds(simplest in some sense). They are homeomorphic to S × (0, 1) where S is a surface of negative Euler characteristics. Quasi-Fuchsian manifolds are naturally compactified by adding two Riemann surfaces S×{0} and S×{1} to M ≃ S×(0, 1). Given a hyperbolic element γ ∈ π1(S) ≃π1(M), we can compute the lengths of the corresponding geodesic in M, S × {0} and S × {1}. In 1970, Bers proved an inequality involving these three lengths which elucidates how conformal boundary can retain information about the hyperbolic manifold.

The goal of this talk will be to state the theorem and to make sense of all the terms in that statement. If time permits, a rough sketch of the proof will be given using an idea of complex analysis, called extremal length.

Nov 13: (No seminar) George Shaji is speaking in the Max Dehn Seminar 

Nov 20: Pratit Goswami (University of Oklahoma)

Title: Stallings' Foldings and the Marshall–Hall Theorem

Abstract:  In 1983, Stallings developed a beautiful way of folding finite graphs for studying finitely generated subgroups of free groups (now commonly known as Stallings' foldings). In this talk, I will introduce the notion of graph immersions and discuss about the Stallings' folding algorithm. I will then present a geometric proof of Marshall-Hall theorem using these folding techniques. We will finally see how all this relate to the notion of 'subgroup separability’.

Nov 27: No seminar (Thanksgiving) 

Dec 4: Jing Tao (University of Oklahoma)

Title: AWM Career Path Talk