Friday, Feb 28, 2025 will be a graduate student workshop, followed by an All-Department Colloquium, hosted by the Student Chapter of the Association for Women in Mathematics at the University of Arkansas.
Conference talks will begin at 8:30am on Saturday, March 1, 2025. All talks are in White Hall, ENGR room 209. Here is a map.
Graduate Student Workshop (Friday)
| Time | Title | Speaker |
|---|---|---|
| 1:00 – 1:50p | Teichmuller Space Invaders | Dan Margalit |
| 2:00 – 2:50p | Birman-Ko-Lee generators | Keiko Kawamuro |
| 2:50 – 3:10p | Break | |
| 3:10 – 4:00p | Thinking about dimension 4 while stuck in dimension 3: Knots, Concordances, and Homology Cobordisms | Miriam Kuzbary |
| 4:30 – 5:30p | Student Chapter of the AWM Colloquium | Siddhi Krishna |
Main Conference
Saturday
| Title | Speaker | |
| 8:30-9:20 | L-spaces and knot traces | John Baldwin |
| 9:30-10:20 | Heegaard Floer homology and the word metric on the Torelli group | Miriam Kuzbary |
| Break | ||
| 10:50-11:40 | Automorphisms of the smooth fine curve graph | Katherine Booth |
| 11:50-12:30 | Lightning Talks: Kailey Perry, Zachary Rail, Brian Udall, Thomas Hill, Shiyu Liang | |
| Lunch | ||
| 2:00-2:50 | The second homology group of the Torelli group | Andy Putman |
| 3:00-3:50 | The contact cut graph and a Weinstein L-invariant | Angela Wu |
| 4:20-5:30 | Lightning Talks: Nilangshu Bhattacharyya, Adithyan Pandikkadan, Jorge Robinson Arrieta, Ethan Dlugie, Rylee Dennis, Rithwik Vidyarthi |
Sunday
| 8:30-9:20 | Mapping Class Groups and Polynomials | Dan Margalit |
| 9:30-10:20 | Taut foliations, braid positivity, and unknot detection | Siddhi Krishna |
| Break | ||
| 10:50-11:40 | Negative band numbers of links and braids | Keiko Kawamuro |
| 11:50-12:40 | How not to study low-dimensional topology? | Michael Klug |
Graduate Workshop Abstracts
Keiko Kawamuro
Birman-Ko-Lee generators
Abstract:In this talk I will advertise the Birman-Ko-Lee generators (also called the band generators) of the braid group. I will demonstrate how to compute the left-canonical form of a braid using hand-written diagrams and give an estimate of the fractional Dehn twist coefficient of the conjugacy class of a braid.
Miriam Kuzbary
Thinking about dimension 4 while stuck in dimension 3: Knots, Concordances, and Homology Cobordisms
Abstract: It is a common theme in topology to study n-manifolds based on the (n-1)-manifolds forming their boundaries, or the (n+1)-manifolds bounded by them. We’ll explore together why this is an interesting and useful thing to do in dimensions 3 and 4! We will talk about knots in the 3-sphere which are secretly related in 4-dimensional ways and how this can help us think about 3-manifolds that are similarly mysteriously connected.
Dan Margalit
Teichmuller Space Invaders
Abstract: What is the most efficient way to transform one Riemann surface into another? What do we mean by efficient? What is a Riemann surface, anyway? We will attempt to answer these questions and more, through a hands-on introduction to these topics. Our secret weapon is an analogy with the singular value decomposition of a matrix. Fortunately, you already internalized the singular value decomposition when you took linear algebra (but just in case, consider brushing up). Our goal is to outline the Bers proof of the Nielsen-Thurston classification theorem, which uses the action of the mapping class group on Teichmuller space.
Main Conference Abstracts
John Baldwin
L-spaces and knot traces
Abstract: There has been a great deal of interest in understanding which knots are characterized by which of their Dehn surgeries. We study a 4-dimensional version of this question: which knots are determined by which of their traces? We prove several results that are in stark contrast with what is known about characterizing surgeries, most notably that the 0-trace detects every L-space knot. Our proof combines tools in Heegaard Floer homology with results about surface homeomorphisms and their dynamics.
Katherine Booth
Automorphisms of the smooth fine curve graph
Abstract: The smooth fine curve graph of a surface is an analogue of the fine curve graph that only contains smooth curves. It is natural to guess that the automorphism group of the smooth fine curve graph is isomorphic to the diffeomorphism group of the surface. But it has recently been shown that this is not the case. In this talk, I will give several more examples with increasingly wild behavior and give a characterization of this automorphism group for the particular case of continuously differentiable curves.
Keiko Kawamuro
Negative band numbers of links and braids.
Abstract: The negative band number is the minimal number of negative bands that are used in braid representatives of a link. It is conjectured that the negative band number is equal to the defect of the Bennequin inequality, another link invariant coming from contact geometry. In this talk I will discuss applications of negative band numbers. Parts of the talk are joint with Michele Capovilla-Serele, Jesse Hamer, Tetsuya Ito, and Rebecca Sorsen.
Michael Klug
How not to study low-dimensional topology?
Abstract: Stallings gave a group-theoretic approach to the 3-dimensional Poincaré conjecture that was later turned into a group-theoretic statement equivalent to the Poincaré conjecture by Jaco and Hempel and then proven by Perelman. Together with Blackwell, Kirby, Longo, and Ruppik, we have extended Stallings’s approach to give group-theoretically defined sets that are in bijection with (i) closed 3-manifolds, (ii) closed 3-manifolds with a link, (iii) closed 4-manifolds, and (iv) closed 4-manifolds with a link (of surfaces). I will explain these bijections and how this results in an algebraic formulation of the unknotting conjecture and a group-theoretic characterization of 4-dimensional knot groups.
Siddhi Krishna
Taut foliations, braid positivity, and unknot detection
Abstract: The L-space conjecture predicts that a manifold with the “extra” geometric structure of a taut foliation also has “extra” Heegaard Floer homological structure. In this talk, I’ll discuss the motivation for this conjecture, describe some results which produce taut foliations in some Dehn surgeries along positive braid knots, and explain how this produces a novel obstruction to braid positivity. Time permitting, I’ll also discuss some work-in-progress with John Baldwin and Matt Hedden, where we obstruct the existence of particular types of taut foliations in some surgeries. I will not assume any background knowledge in Floer or foliation theories; all are welcome!
Miriam Kuzbary
Heegaard Floer homology and the word metric on the Torelli group
Abstract: Since its inception, Heegaard Floer homology has been an invaluable tool for the study of 3- and 4- manifolds. Diffeomorphisms of surfaces (up to isotopy) are inherent to its construction; however, more remains to be explored about how invariants in Heegaard Floer homology interact with techniques and structures in mapping class groups of surfaces. In this talk, I will discuss recent work with Santana Afton and Tye Lidman where we study the relationship between the Heegaard Floer homology correction terms of integral homology spheres and the word metric on the Torelli group and we discover some surprising phenomena.
Dan Margalit
Mapping Class Groups and Polynomials
Abstract: A polynomial in one complex variable gives a branched cover of the Riemann sphere. But which branched covers of the Riemann sphere come from polynomials? In the early 1980s, William Thurston gave one answer to this question: the “obvious” obstruction is the only obstruction. Shortly thereafter, Hubbard asked his famous twisted rabbit problem: if we post-compose a particular polynomial by a homeomorphism, which polynomial is obtained, if any? A solution to this was found by Bartholdi and Nekrashevych in 2006. In this talk, I will give an introduction to this circle of ideas, highlighting recent work that draws strong parallels with the theory of mapping class groups of surfaces.
Andy Putman
The second homology group of the Torelli group
Abstract: I will explain how to calculate the second rational homology group of
the Torelli group. This is joint with Dan Minahan.
Angela Wu
The contact cut graph and a Weinstein L-invariant
The cut complex associated to a surface is a powerful tool in the study of smooth 4-manifolds. In this talk, I will introduce the cut complex and its analogue in the contact and symplectic setting. Adding a restriction to the allowable curves in a cut system, the contact cut graph is a subgraph of the cut complex. I will go through some recent results exploring the structure of the contact cut graph, and use it to define a new invariant of 4-dimensional Weinstein domains. This is based on joint work with Castro, Islambouli, Min, Sakalli, and Starkston.