Date: Tuesday Sept 15, 2026
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Polyhedral Decompositions of Knot Complements
Speaker: Anay Jain
Abstract: I will motivate the study of the geometry of knot complements. One interesting combinatorial way to construct a hyperbolic structure on the complement of a knot is to decompose it into two polyhedra. I will illustrate this construction for the figure-8 knot, and if time permits, I will talk about the conditions for when this decomposition admits a geometric structure. This talk will be have no prerequisites, but basic knowledge of Riemannian geometry and Emma's wonderful talk last week will be insightful.
Date: Tuesday Sept 22, 2026
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Fox derivatives and the Alexander polynomial
Speaker: Éamonn Olive
Abstract: As an introduction to the topic, we will show how you could accidentally invent the Fox derivative to solve problems in group theory. We will then show surprising connections to knot theory and the Alexander polynomial. This will serve as an introduction to both the Fox derivative and the Alexander polynomial; I will not expect foreknowledge in either.
Date: Tuesday Sept 29, 2026
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: The Alexander polynomial from Fox derivatives
Speaker: Éamonn Olive
Abstract: We will introduce knot groups and their basic properties. From there we will use the Fox derivative to derive the Alexander module and polynomial from the knot group. This is the second part of a two-part talk. The first part dealt mainly with the motivations behind the Fox derivative. In order to make this talk legible to people who missed the first, we will recall its definition and some key facts.
Date: Tuesday Oct 6, 2026
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Mostow Rigidity
Speaker: Summer Eldridge
Abstract: The geometry and topology of a hyperbolic 3-manifold is uniquely determined by its fundamental group; This result undergirds any study of hyperbolic 3-manifolds. I'll be walking you through proof in Martelli's Geometric Topology. Some background in algebraic topology is required, and a little bit in hyperbolic geometry, but the talk is mostly self-contained and requires almost no advanced machinery.
Date: Tuesday Oct 20, 2026
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Gromov Boundary of the Grand Arc Graph
Speaker: Arya Vadnere (SUNY Buffalo)
Abstract: In 1999, E. Klarreich found a very intriguing correspondence between the Gromov boundary of the curve graph for closed surfaces (a very GGT object) with the space of ending laminations on the surface (a very geometric object). Since then, Hamendstadt, Schleimer and Pho-On have thought about various proofs for this result, and generalizations to the arc graph / the arc-and-curve graph for finite-type surfaces. The grand arc graph is a type of arc graph associated with certain infinite-type surfaces, which is also an infinite-diameter hyperbolic graph. In this talk, we shall talk about a couple of ways to define "laminations that should correspond to points on the Gromov boundary of the grand arc graph". This work is joint with Carolyn Abbott and Assaf Bar-Natan.