Research Seminars: Geometry & Topology

Sping 2024

Time & Location: All talks are on Monday in Gibson Hall 308 at 2:00 PM unless otherwise noted.
Organizer: Komendarczyk, Rafal



March 4

Title: Knot Invariants, Categorification, and Representation Theory

Arik Wilbert - University of South Alabama

Abstract: I will provide a survey highlighting connections between representation theory, low-dimensional topology, and algebraic geometry central to my research. I will recall basic facts about the representation theory of the Lie algebra sl2 and discuss how these relate to the construction of knot invariants such as the well-known Jones polynomial. I will then introduce certain algebraic varieties called Springer fibers and explain how they can be used to geometrically construct and classify irreducible representations of the symmetric group. These two topics turn out to be intimately related. More precisely, I will demonstrate how one can study the topology of certain Springer fibers using the combinatorics underlying the representation theory of sl2. On the other hand, I will show how Springer fibers can be used to categorify certain representations of sl2. As an application, one can upgrade the Jones polynomial to a homological invariant which distinguishes more knots than the polynomial invariant. Time permitting, I will discuss how this picture might generalize to other Lie types beyond sl2.

Location: Gibson Hall 308

Time: 2:00


April 12

Title: Knot invariants and hyperbolic flows

Solly Coles - Northwestern University

Abstract: In this talk, we will discuss the average value taken by a knot invariant on the periodic orbits of a hyperbolic flow on the 3-sphere. 
The first relevant result comes from the work of Contreras, who studied the average linking number between periodic orbits. Contreras found precise asymptotic growth rates for this number, as the period tends to infinity. In the proof, the Gauss linking integral is used to translate the problem into the language of ergodic theory.
In recent work, we instead consider the average value of a Vassiliev invariant on periodic orbits. Here, the configuration space integrals of Bott and Taubes take the place of the Gauss linking integral in Contreras' work.

Location: Gibson Hall 308

Time: 2:00


April 22

Title: The wrappingness and trunkenness of volume-preserving flows

Peter Lambert-Cole - University of Georgia

Abstract: Link invariants of long pieces of orbits of a volume-preserving flow can be used to define diffeomorphism invariants of the flow.  The wrapping number of a link in the solid torus and the trunk of a link can be generalized and define invariants of links with respect to a fibration on a 3-manifold. Extending work of Dehornoy and Rechtman, we apply this to define diffeomorphism invariants wrappingness and trunkenness of volume-preserving flows on 3-manifolds and interpret these invariants as obstructions to the existence of a global surface of section for the flow. We construct flows and show that wrappingness and trunkenness are not functions of the helicity.

Location: Gibson Hall 308

Time: 2:00


April 25

Title: Distortion and Curvature in the Shape Reconstruction Problem

Will Lopez Tran | Tulane University

Abstract: We consider the complete topological reconstruction of a geodesic subspace of Euclidean space from the Vietoris--Rips complexes of a finite, noisy Euclidean sample. Instead of the Euclidean metric, our reconstruction technique uses a path-based metric on the sample to construct the Vietoris--Rips complexes. 

We consider the restricted distortion, alpha-distortion, convexity radius, and Alexandrov curvature of a geodesic space as our sampling parameters. With restricted distortion, we guarantee a homotopy-equivalent reconstruction from the sample. With alpha-distortion, we guarantee homology groups equivalence and fundamental  group equivlance with reconstruction from the sample. This study provides alternative sampling conditions to the existing and commonly used conditions based on weak feature size and $\mu$--reach.

Location: Newcomb 411 or zoom:

Time: 10:00am