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

Title: They Did Surgery on a Four Manifold

Abstract: Cut and paste surgery techniques are a common and successful method of constructing low dimensional manifolds. For example, in three dimensions, the Lickorish-Wallace theorem states that any (closed, connected, orientable) 3-manifold can be obtained by performing ±1 Dehn surgery on a framed link in S3. In this talk, we will discuss two 4-dimensional analogs of Dehn surgery: the Log Transform and Fintushel-Stern's knot surgery. Using knot surgery, we will construct an example of two homeomorphic four manifolds which are not diffeomorphic. 

10/1 –  Jen Rozenblit

Title:  Graph Cobordisms and Neural Networks  

Abstract: Graph Neural Networks (GNNs) have become a central tool in modern machine learning, but their algebraic and topological structure is often hidden under heuristic descriptions. In this talk, we reframe GNNs in a language more familiar to us fellow topologists: cobordisms and functors between categories. We introduce a category of graph cobordisms (GCob), where morphisms are open graph cobordisms encoding how graph topology can evolve layer by layer. In parallel, we build the analytic counterpart EucLip, whose morphisms are Lipschitz maps between Euclidean block spaces, capturing quantitative notions such as stability and sensitivity. A central theme is the functor F: GCob→EucLip, which translates combinatorial gluing of graphs into analytic composition of GNN layers. We discuss the purpose/construction of this F.

No prior knowledge of AI, neural networks, or GNNs is assumed — all necessary definitions will be given, and the talk will be self-contained. This is ongoing research - still very much in progress - and I will emphasize both the constructions so far and the open directions that remain. 

10/8 – Justin Chen

Title: The barcode of a Morse function

Abstract: What can the critical values of a smooth function on a manifold M tell us about where the function lies in C^infty(M)? Not much by themselves – by introducing small oscillations in a function, we can always introduce new critical values far away from the original ones. To remedy this, we borrow ideas from Morse theory and topological data analysis to organize the critical values of a function into a combinatorial object called a barcode. We will show how the space of barcodes reflects the space of Morse functions on M in a geometric way.

10/22 – Aru Mukherjea

Title: Extendible mapping classes of surfaces in 4-manifolds

Abstract: Knotted surfaces are defined as embeddings of surfaces into 4-manifolds, but our mental picture is usually of the image of this embedding, up to isotopy. A natural question, then, is how many isotopy classes of embeddings have the same image? In this talk we explore this by introducing the notion of extendible mapping classes of surfaces in 4-manifolds and some constructions and obstructions to finding them. As an application, we give an alternative proof that every 2-knot is slice.

10/29 – Hillary Kim

Title: Knot Concordance Group and Hedden's Conjecture

Abstract: We will go over the knot concordance group as well as the basic definitions of patterns and satellite knots. We will also talk about Hedden's Conjecture as well as a small proof as part of its partial resolution using branched cover obstructions

11/5 – John Teague

Title: Studying Satellites and Concordance with Instantons

Abstract: Last week, Hillary told us about the relationship between satellite knots and concordance groups, mentioning a conjecture of Hedden asserting that satellite operations almost never induce homomorphisms. In this talk, I’ll investigate a slightly different conjecture regarding the rank of satellite maps (also due to Hedden). Using a clever application of Chern-Simons gauge theory, we’ll show that a large class of satellite maps have images which are of infinite rank over Z. In special situations we’ll get even more: satellite maps which are zero on the topological concordance group (i.e. have images consisting of topologically slice knots) but are infinite rank on the smooth concordance group. No gauge theory background is required! But I will make brief comparisons with Seiberg-Witten theory.

P.S. These results are from the Hedden-Pinzón-Caicedo paper that Ivan So mentioned in the topology seminar!

11/12 – Peter Seong

11/19 – Audrick Pyronneau

12/3 – TBA

Schedule TBA