Schedule, Titles, Abstracts

Saturday:

8:00-9:00 coffee/breakfast

9:00-10:00 Krushkal

10:15-11:15 Miller

11:30-12:30 Munoz-Echaniz

Lunch

2:00-3:00 Willis

3-3:30 Coffee

3:30-4:30 Zhang

Sunday:

8:30-9:30 coffee/breakfast

9:30-10:30 Min

11:00-12:00 Shen

Lunch

1:30-2:30 Hendricks

2:30-3 Coffee

3:00-4:00 Hartman

Monday:

8:30-9:30 coffee/breakfast

9:30-10:30 Stoffregen

11:00-12:00 Li

Lunch

1:30-2:30 Ruberman

2:30-3 Coffee

3:00-4:00 Khan

Daniel Harold Hartman

pseudo-isotopies of simply connected 4-manifolds.

Published in 1986, Quinn provided a proof that for a simply-connected 4-manifold, any homeomorphism pseudo-isotopic to the identity is actually isotopic to the identity. Additionally for smooth manifolds, diffeomorphisms that are pseudo-isotopic are smoothly stably isotopic. Both of these results are fundamental to the world of 4-manifolds. Part of the strategy Quinn's employs for deriving these results revolves reducing the problem down to understanding and manipulating a specific collection of embedded disks. There is one particular "move" that is critical to Quinn's argument for both the smooth stably isotopic and topological pseudo-isotopy implies isotopy: the replacement criterion. However, the justification for using this move is incorrect. I plan to discuss the replacement criterion and give a way to circumvent it in order to complete Quinn's proofs.

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.

Isabella Khan

Dual algebras in bordered Heegaard Floer homology

By slicing the Heegaard diagram for a given 3-manifold in a particular way, it is possible to construct $\A_{\infty}$-bimodules, the tensor product of which retrieves the Heegaard Floer homology of the original 3-manifold. The first step in this is to construct algebras corresponding to the individual slices. This talk will outline the construction of two Koszul dual $\A_{\infty}$ algebras, $\mathcal{A}$ and $\mathcal{B}$, for a particular star-shaped class of slice, as well as the verification of the Koszul duality relation.

Slava Krushkal

Using pseudoisotopy theory to localize exotic diffeomorphisms

Exotic smooth structures on simply-connected 4-manifolds are known to be related by cork twists: cutting out and re-gluing certain smooth contractible submanifolds. Work in progress, joint with Anubhav Mukherjee, Mark Powell, and Terrin Warren, provides a localization result for exotic diffeomorphisms of 4-manifolds. I will also discuss applications to known examples of exotic diffeomorphisms. 

Zhenkun Li

Instanton Floer homology and Heegaard diagrams

Floer theory is a power 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 first present a result joint with Baldwin and Ye that illuminates some of the interplay between these two theories. Time permitted, I will delve into ongoing research that further investigates these intriguing connections

Allison Miller

A partial resolution of Hedden's conjecture

A pattern, or knot in a solid torus, induces a map on the set of knots modulo smooth concordance. In 2016, Hedden conjectured that essentially none of these maps are group homomorphisms--more precisely, that the only homomorphisms induced by patterns are the identity map, the reversal map, and the zero map. In particular, this would imply that patterns with winding number not in the set {-1,0,1} definitely do not induce homomorphisms. I will discuss work with Randall Johanningsmeier and Hillary Kim in which we prove that if P is a pattern with winding number that is even but not divisible by eight, then P cannot induce a homomorphism on the smooth concordance group. This relies heavily on previous joint work with Tye Lidman and Juanita Pinzon-Caicedo, and is the first result that obstructs all patterns of a fixed winding number from inducing homomorphisms.

Hyunki Min

Tight contact structures on hyperbolic L-spaces

The classification of contact structures is one of the interesting problems in low-dimensional contact topology. In this talk, we will give an overview of the classification of tight contact structures. We will first discuss the general strategy and the classification of tight contact structures on Seifert fibered spaces. After that, we will talk about the classification of tight contact structures on hyperbolic 3-manifolds, particularly hyperbolic L-spaces. This is a joint work with Isacco Nonino.

Juan Munoz-Echaniz

Dehn twists on 4-manifolds with Seifert-fibered boundary

A smooth 4-manifold with boundary on a Seifert-fibered 3-manifold admits a "Dehn twist" diffeomorphism supported inside a collar neighbourhood of its boundary. Such diffeomorphisms naturally arise, for instance, as the monodromy of surface singularities. By results of Orson--Powell, these Dehn twists are very often topologically isotopic to the identity. We show that these Dehn twists are not smoothly isotopic to the identity and have infinite order in the case of symplectic fillings with b^+ > 0 of several Seifert-fibered 3-manifolds. This talk is based on joint work in progress with Hokuto Konno, Jianfeng Lin and Anubhav Mukherjee.

Danny Ruberman

Exotic aspherical 4-manifolds

Abstract: We construct smooth aspherical 4-manifolds M and M’ along with a homeomorphism between them that is not homotopic to a diffeomorphism.  This is ongoing joint work with Kyle Hayden, Lisa Piccirillo, and Nathan Sunukjian.

Weizhe Shen

Persistent Legendrian contact homology

Persistent homology is a technique widely used in topological data analysis. Legendrian contact homology (LCH) is a powerful tool to use on the classification problem in Legendrian knot theory. In this talk, we will discuss how to introduce persistent homology computations to LCH and some results that we get by applying the idea of persistence. We will then describe some ongoing research that further investigates the theory. This is joint work with Maya Basu, Austin Christian, Ethan Clayton, Daniel Irvine, and Fredrick Mooers.

Matt Stoffregen

Lattice Floer Spectra

Recently, Zemke proved that Heegaard Floer homology and lattice homology agree, for general plumbing trees, generalizing a theorem of Ozsváth-Szabó showing this equivalence for almost-rational plumbings.  In this talk, we'll review some facts about monopole Floer spectra, and give a calculation of the monopole Floer spectra of almost rational plumbings, based closely on Ozsváth-Szabó's proof, in terms of lattice homology.  We also include some obstructions to the existence of spin 4-manifolds with certain boundary that follow from these calculations.  This is joint work with Irving Dai and Hirofumi Sasahira.

Mike Willis

Khovanov skein lasagna modules and exotica

The Khovanov skein lasagna module S(X;L) is a smooth invariant of a 4-manifold X with link L in its boundary.  In this talk I will outline the construction of Khovanov skein lasagna modules, as well as new computations and applications including the detection of some exotic 4-manifolds.  This work is joint with Qiuyu Ren.

Melissa Zhang

Kirby and the Skein Lasagna Module of $S^2 \times S^2$

In 2018, Morrison, Walker, and Wedrich’s skein lasagna modules are 4-manifold invariants defined using Khovanov-Rozansky homology similarly to how skein modules for 3-manifolds are defined. In 2020, Manolescu and Neithalath developed a formula for computing this invariant for 2-handlebodies by defining an isomorphic object called cabled Khovanov-Rozansky homology; this is computed as a colimit of cables of the attaching link in the Kirby diagram of the 4-manifold.

In joint work with Ian Sullivan, we lift the Manolescu-Neithalath construction to the level of Bar-Natan's tangles and cobordisms, and trade colimits of vector spaces for a homotopy colimit in Bar-Natan's category. As an application, we give a proof that the skein lasagna module of $S^2 \times S^2$ is trivial, confirming a conjecture of Manolescu. Our local techniques also allow for computations of the skein lasagna invariant for other 4-manifolds whose Kirby diagram contains a 0-framed unknot component. Our methods also allow us to relate the Rozansky-Willis invariant of links in $S^2 \times S^1$ to skein lasagna modules.