UT Junior Topology Seminar

The UT Junior Topology Seminar is run by and for graduate students interested in topology and geometric structures. Weekly talks are given by grads at all stages of the PhD program, on topics including introductions to foundational tools, exciting results from recent (or not-so-recent) papers, and students' original research. Speakers have the opportunity to gather constructive feedback after each talk.

9/3 – Aaron Benda

Title: The Nielsen Realization Theorem

Abstract: Surfaces, the younger sibling to 3 and 4 manifolds, exhibit extremely rich topological and geometric features. We will begin by reviewing the basic topology of surfaces and hyperbolic geometry, before introducing the main player for this talk: the mapping class group. Mapping classes are isotopy classes of self-homeomorphisms, and this collection forms a group associated to a surface with an often rich structure. We will cover some examples before asking the main question, linking the geometry and topology of surfaces: when do finite groups of mapping classes arise from groups of isometries for a hyperbolic metric? This question (initially posed in the 1930s) was once known as the Nielsen Realization Problem, and was resolved by Kerckhoff in 1983. We won't cover the proof, which relies on deep facts about Teichmuller space and deformations of hyperbolic structures called earthquake paths, but we will state and break down the result.

9/10 – Remy Bohm

Title: Equivariant concordance of periodic 2-knots

Abstract: We show that the equivariant concordance group of smooth 2-knots in S4 invariant under a Z/dZ action, where the action is given by rotation about an unknotted sphere intersecting the 2-knot in two points, is isomorphic to Z/2Z for all d ≥ 2. This is in contrast to the non-equivariant case, in which all 2-knots are slice. We construct a new invariant for these 2-knots, which we call periodic, and show that it fully classifies them up to equivariant concordance. The invariant depends on an extension of the Arf invariant for null-homologous classical knots in arbitrary 3-manifolds.

9/17 – Mark Saving

Title: Factorization Homology

Abstract: We all know and love (extraordinary) homology theories, which we can broadly describe as functors that produce chain complexes from spaces and satisfy certain axioms. We introduce factorization homology, a powerful machine which takes an algebra and produces a homology theory for manifolds. We discuss how this tool can be used to study both algebras and manifolds, recovering well-known invariants like Hochschild and singular homology. Time permitting, we discuss other applications of the theory, including its relation to skein theory and TQFTs.

9/24 – Jemma Schroder

10/1 –  Jen Rozenblit

10/8 – Justin Chen

10/22 – Aru Mukherjea

10/29 – John Teague

11/5 – Hillary Kim

11/12 – Peter Seong

11/19 – Audrick Pyronneau

12/3 – TBA

Schedule TBA