2024 Geometry and Topology Workshop at UCLA

Date: January 4-6, 2024
Location: IPAM, UCLA
Organisers: Ko Honda and Sucharit Sarkar
Funded by: NSF DMS-2136090
With support from: UCLA Mathematics Department and IPAM

The workshop focused on recent developments in contact geometry and low-dimensional topology, with a particular emphasis on modern homological invariants. The workshop was open to all; the following photo captures most of the participants.

Schedule

All talks were held at the Institute for Pure and Applied Mathematics (map of its location). Select a talk title to show or hide its abstract.

Thursday, January 4, 2024

09:00-10:00: Breakfast
10:00-11:00: Abhishek Mallick
Gompf's cork and Heegaard Floer homology

Corks are fundamental to the study of exotic smooth structures on 4-manifolds. In this talk we will discuss how to use Heegaard Floer homology to detect corks. We will then describe some new examples of corks which partially address a question posed by Gompf. This is joint with Irving Dai and Ian Zemke.
11:15-12:15: Jonathan Hanselman
Satellite knots and immersed curves

Satellite operations are a valuable method of constructing complicated knots from simpler ones, and much work has gone into understanding how various knot invariants change under these operations. We describe a new way of computing the (UV=0 quotient of the) knot Floer complex using an immersed Heegaard diagram obtained from a Heegaard diagram for the pattern and the immersed curve representing the UV=0 knot Floer complex of the companion. This is particularly useful for (1,1)-patterns, since in this case the resulting immersed diagram is genus one and the computation is combinatorial. In the case of one-bridge braid satellites the immersed curve invariant for the satellite can be obtained directly from that of the companion by deforming the diagram, generalizing earlier work with Watson on cables. This is joint work with Wenzhao Chen.
12:15-14:15: Lunch break
14:15-15:15: Zhenkun Li
Instanton Floer homology and Heegaard diagrams

Floer theory is a powerful tool in the study of low dimensional topology, leading to many milestone results in the field. There are four major branches of Floer homologies, all of which have distinct features and applications. Among them, Heegaard Floer homology, monopole Floer homology, and embedded contact homology are known to be isomorphic, yet their relationship with Instanton Floer homology remains enigmatic. This talk will explore the connection between Instanton and Heegaard Floer homology. I will present a result joint with Baldwin and Ye that illuminates the interplay between these two theories. Time permitting, I will delve into ongoing research that further investigates these intriguing connections.
15:30-16:30: Akram Alishahi
Bordered Floer homology and compressing surface diffeomorphisms

Let $F$ be a closed surface, and $φ$ be a diffeomorphism of $F$ . An interesting question with nice topological implications for detecting homotopy ribbon fibered knots is whether $φ$ extends over some handlebody with boundary $F$ . In 1985, Casson-Long gave an algorithm for answering this question. In this talk, we will discuss answering this question using bordered Floer homology. First, we will talk about how bordered Floer homology can be used to detect whether $φ$ extends over a specific handlebody. Then, we will outline how to adapt ideas of Casson-Long to give an algorithm using bordered Floer homology to detect whether a mapping class extends over any compression body. This is a work in progress with Robert Lipshitz.

Friday, January 5, 2024

09:00-10:00: Breakfast
10:00-11:00: Jennifer Hom
Satellite operations and complexity

How do various notions of knot complexity behave under satellite operations? In many cases, there are "obvious" upper bounds and one can ask when such bounds are in fact optimal. We will consider unknotting number and fusion number under the operation of cabling. This is joint work with Sungkyung Kang, Tye Lidman, and JungHwan Park.
11:15-12:15: Maggie Miller
Fibered(?) ribbon knots

In 1983, Casson-Gordon showed how to use fibered ribbon knots to construct potentially exotic homotopy 4-balls. They did this by constructing a disk bounded by such a knot into some homotopy 4-ball, extending the fibration from the knot complement to the disk complement. In this talk, I will explain this setup and potential consequences with respect to several open problems in low-dimensional topology, and my work from 2018, 2020 (with A. Zupan), 2023 and the future on controlling the extension of these fibrations (or generalizations thereof). Some of the work in this talk is joint with Alex Zupan.
12:15-14:15: Lunch break
14:15-15:15: Anubhav Mukherjee
An eye towards understanding of smooth mapping class group of 4-manifolds

The groundbreaking research by Freedman, Kreck, Perron, and Quinn provided valuable insights into the topological mapping class group of closed simply connected 4-manifolds. However, the development of gauge theory revealed the exotic nature of the smooth mapping class group of 4-manifolds in general. While gauge theory can at times find the existence of smooth isotopy between two diffeomorphisms, it falls short of offering a comprehensive understanding of the existence of diffeomorphisms that are topologically isotopic but not smoothly so. In this talk, I will elucidate some fundamental principles and delve into the origins of such exotic diffeomorphisms. This is my upcoming work joint with Slava Krushkal, Mark Powell, and Terrin Warren.
15:30-16:30: Laura Starkston
Encoding Weinstein structures with sequences of Heegaard diagrams

We will talk about Heegaard diagrammatic ways to encode Weinstein 4-manifolds (symplectic manifolds with compatible handle structures). This is based on joint work with Gabriel Islambouli.

Saturday, January 6, 2024

09:00-10:00: Breakfast
10:00-11:00: Orsola Capovilla-Searle
Results on exact Lagrangian fillings of Legendrian links

An important problem in contact topology is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Legendrian links can also arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4-ball. Such surfaces are called fillings of the link. In the last decade, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. I will talk about new connections between fillings and Newton polytopes, as well as results on distinguishing non-orientable fillings. This is based on joint work with Casals and joint work with Hughes and Weng.
11:15-12:15: Aliakbar Daemi
Rank three instantons, representations and sutures

Yang-Mills gauge theory with gauge group SU(2) has played a significant role in the study of the topology of 3- and 4-manifolds. It is natural to ask whether we obtain more topological information by working with other choices of gauge groups such as SU( $n$ ) for higher values of $n$ . Mariño and Moore formulated a conjecture essentially stating that there is no new information in Donaldson invariants of smooth 4-manifolds defined using SU( $n$ ) Yang-Mills gauge theory. Despite this "negative" prediction, one might still hope that there is still novel information about 3-manifolds in higher rank gauge theory. In this talk, I will discuss a result about the topology of 3-manifolds obtained using gauge theory with respect to the Lie group SU(3): for any knot $K$ in the 3-dimensional sphere (or more generally an integer homology sphere) there is a non-abelian representation of the knot group of $K$ into SU(3) such that the homotopy class of the meridian of $K$ is mapped to a matrix with eigenvalues $1, w, w^{2}$ with $w$ being a primitive third root of unity. As a byproduct of the proof, we obtain a structure theorem for SU(3) Donaldson invariants of 4-manifolds, analogous to Kronheimer and Mrowka's structure theorem for SU(2) Donaldson invariants. This can be regarded as a piece of evidence supporting Mariño and Moore's conjecture. This talk is based on a recent joint work with Nobuo Iida and Chris Scaduto.
12:15-14:15: Lunch break
14:15-15:15: Irving Dai
Involutions and the Chern-Simons filtration in instanton Floer homology

Building on the work of Nozaki, Sato and Taniguchi, we develop an instanton-theoretic invariant aimed at studying strong corks and equivariant bounding. As an application, we give an example of a cork whose boundary involution does not extend over any 4-manifold $X$ with $H_{1} (X, ℤ_{2}) = 0$ and $b_{2} (X) \leq 1$ , and a strong cork which survives stabilization by either of ${ℂ P}^{2}$ or ${\bar{ℂ P}}^{2}$ . We also discuss applications to the geography question for nonorientable surfaces in the case of extremal normal Euler number. This is joint work with Antonio Alfieri, Abhishek Mallick, and Masaki Taniguchi.
15:30-16:30: Kristen Hendricks
Symplectic annular Khovanov homology and knot symmetry

Khovanov homology is a combinatorially-defined invariant which has proved to contain a wealth of geometric information. In 2006 Seidel and Smith introduced a candidate Lagrangian Floer analog of the theory, which has been shown by Abouzaid and Smith to be isomorphic to the original theory over fields of characteristic zero. The relationship between the theories is still unknown over other fields. In 2010 Seidel and Smith showed there is a spectral sequence relating the symplectic Khovanov homology of a two-periodic knot to the symplectic Khovanov homology of its quotient; in contrast, in 2018 Stoffregen and Zhang used the Khovanov homotopy type to show that there is a spectral sequence from the combinatorial Khovanov homology of a two-periodic knot to the annular Khovanov homology of its quotient. (An alternate proof of this result was subsequently given by Borodzik, Politarczyk, and Silvero.) These results necessarily use coefficients in the field of two elements. This inspired investigations of Mak and Seidel into an annular version of symplectic Khovanov homology, which they defined over characteristic zero. In this talk we introduce a new, conceptually straightforward, formulation of symplectic annular Khovanov homology, defined over any field. Using this theory, we show how to recover the Stoffregen-Zhang spectral sequence on the symplectic side. We further give an analog of recent results of Lipshitz and Sarkar for the Khovanov homology of strongly invertible knots. This is work in progress with Cheuk Yu Mak and Sriram Raghunath.