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.

1/28 – Laura Wakelin*

Title: JSJ decompositions of knot exteriors. 

Abstract: I will give an overview of JSJ decompositions of knot exteriors, as explained in Budney's survey paper. I will then explain how such a JSJ decomposition can change after Dehn filling, which is a useful tool for answering questions about (non-)characterising slopes and cosmetic surgeries. 

*postdoc

2/4 – Hyunsoo Park^

Title:  Local Equivalences in Heegaard Floer theory

Abstract: Heegaard Floer homology, introduced by Ozsváth and Szabó, associates to a 3-manifold a chain complex over a polynomial ring, equipped with rich algebraic structures. In this talk, I will begin with a brief review of the formal algebraic structure of Heegaard Floer homology, focusing on the d-invariant. I will then introduce the notion of local equivalence, which is an equivalence relation reflecting how Floer complexes behave under maps induced by homology cobordisms. Finally, I will discuss applications of local equivalence to low-dimensional topology, especially in the study of the knot concordance group and the homology cobordism group.

^undergraduate

2/11 – Angel Yuan^

Title: Spectra and representability, part 2

Abstract: I will roughly outline the framework for the proof of representability of generalised homology theories. We will review the basics of spectra and talk about finite spectra (spectra of finite CW complexes) and the Spanier-Whitehead category (the symmetric monoidal category they live in). This is a standalone sequel to my talk from last semester.

^undergraduate

2/18 – Mark Saving

Title: Homotopy Type Theory: The Next Great Foundation of Math?

Abstract: We introduce Homotopy Type Theory (HoTT), a new foundation of mathematics developed in the 21st century that makes homotopy types the basic objects of the theory and forces all constructions to be homotopy-coherent automatically and painlessly. We discuss some of the achievements and limitations of HoTT, including whether it can fully replace ZFC as the dominant foundational theory.

2/25 – Matt Elpers†

Title:  Non-characterizing slopes for more 3-manifolds

Abstract: Let M be any compact, connected, orientable 3-manifold, such that M is closed or ∂M contains only tori. For every homotopy class h in π1(M) there are infinitely many pairs of knots representing h which have orientation-preserving homeomorphic 0-surgeries.

† invited speaker from NC State University 

3/4 – Ansel Goh

Title: h-cobordisms between 1-connected manifolds with boundary

Abstract: I’ll be talking about recent work classifying h-cobordisms between simply connected 4-manifolds with boundary up to homeomorphism/diffeomorphism rel boundary. This extends Kreck’s classification of h-cobordisms between closed simply connected 4-manifolds. As an easy consequence, we also obtain an alternative proof for the injectivity portion of Orson-Powell’s computation of the mapping class group for simply connected topological 4-manifolds with boundary.

3/11 – Ioannis Karagiorgis

Title: Can we prove the Four Colour Theorem using instanton Floer homology?

Abstract: The Four Colour Theorem was first proven in 1976 by Appel and Haken, in what is widely regarded as the first computer-assisted proof of a major theorem. The numerous reworked proofs since then all crucially require computer assistance. But can we do better and find a fully human-verifiable proof? In 2015, Kronheimer and Mrowka used SO(3) gauge theory to define an instanton invariant for webs (embedded trivalent graphs in R^3) that counts certain representations of π_1 of the web complement into SO(3). In the case of planar webs, they conjectured that their invariant counts the number of Tait colourings of the web. I will give an overview of the construction of this invariant and illustrate how their conjecture, if true, would lead to a new, computer-free proof of the Four Colour Theorem.

3/25 – Peter Seong

Title: Finite Gauge Theory

Abstract: Gauge theory is quite analysis-heavy. It tells us a lot of things about physics and smooth topology.

Can you believe if there is a way to do Gauge theory completely analysis-free?

We will set a structure group to be any finite group. Surprisingly, it still exhibits the interesting algebraic and topological structures. I may define some “quantum” invariants for links on the finite setting and  show that they work(!) (i.e. baby link invariants) if time permits.

4/1 – Hillary Kim

Title: A cute proof on why the Figure 8 knot is not slice 

Abstract: I will do what the title says and also talk about why the 2,1 cable of the figure 8 is not smoothly slice!

4/8 – Justin Chen

Title: Group trisections of 4-manifolds

Abstract: In this talk we’ll introduce the notion of a trisection of a 4-manifold, a dimension 4 extension of Heegaard splittings for 3-manifolds. We’ll see how the data of a smooth 4-manifold can be expressed in terms of Heegaard diagrams, which can in turn be captured by maps on fundamental groups. This will allow us to rephrase questions in 4-manifold topology, such as the smooth Poincare conjecture, into questions of group theory.

4/15 – Nathan Louie

Title: It's Stonging time

Abstract: This week, I will discuss the Stong invariant, a secondary invariant to the Freedman–Quinn invariant in the problem of upgrading homotopies of 2-spheres with geometric duals to concordances (see Maggie’s 4/4 class). I will give a brief sketch of the operations and constructions Stong uses to define this invariant.

4/22 – Elise Brod

Title: Using Heegaard diagrams to construct 4-manifolds

Abstract: Given a closed orientable 3-manifold Y, and a smooth, compact, orientable 4-manifold filling X such that ∂X=Y, we might wonder what X could be (e.g. exploring which rational homology 3-spheres bound rational homology 4-balls). In this expository talk, we will discuss a hands-on strategy for producing fillings of Y. We will start with a a standard Heedaard diagram for #kS1 x S2 and use d Dehn twists to modify it until we arrive at a Heegaard diagram for Y. Using the Lickorish Trick and that fact that ∂({0-handle} ∪{k 1-handles}) =  #kS1 x S2, we construct X from a single 0-handle, k 1-handles, and d 2-handles, where the 2-handle attaching instructions are explicitly determined by our Dehn twists.

The focus will be on small visual examples, but based on audience interest I might (1) discuss some applications (e.g. to embedding problems), (2) convince you that this strategy is useful by sketching the proof that all fillings* for Y can be constructed this way, or (3) convince you that this strategy is not useful by showing you why some enticing applications of this strategy actually boil down to solving the word problem for the mapping class group of a genus g surface (hard).

*using 3-handles requires more work

4/29 – Adrian Flores

Title: Supersymmetric Yang--Mills and the Seiberg--Witten equations

Abstract: I will try to introduce the physical ideas that led to the monopole equations