Talks

Talks AY 2025-2026

None currently scheduled, but here is a tentative plan for talks as the Minerva Visitor at Princeton in Fall 2025.

Talks AY 2024-2025

Towards a Legendrian contact stable homotopy type

September 12, 2024 in the Princeton Topology Seminar.

Abstract:

The Chekanov-Eliashberg dga, or Legendrian contact homology, was the first modern invariant of Legendrian knots in $\mathbb{R}^3$. This talk is a progress report on a project to give a stable homotopy refinement of Legendrian contact homology, inducing operations like Steenrod squares on linearized Legendrian contact homology. In general, contact homology is defined by counting psuedoholomorphic curves, but in this case those counts reduce to the Riemann Mapping Theorem and the invariant is described purely combinatorially, from an appropriate knot diagram, and our refinement has a similarly combinatorial flavor. The talk will start at the beginning, briefly recalling the basics of Legendrian knot theory and the construction of Legendrian contact homology, and then outline our program to refine it, sketch its status, and describe some examples. This is joint with Lenhard Ng and Sucharit Sarkar. LCH

Exotic Surfaces

October 3, 2024 in the UIUC Mathematics Department Colloquium.

Abstract:

In the last few years there has been an explosion of new results about surfaces in 4-space. In this talk, we will start by discussing various kinds of surfaces and some basic questions about them, like what it means for two of them to be the equivalent. We will then discuss two ways to describe smooth surfaces in 4-space explicitly, give a number of examples illustrating recent results, and mention some still-open questions. The talk will end with a sketch of some of the techniques involved. Most of the results in the talk are due to other people; a few are joint with Sucharit Sarkar.Colloquium / Overview

Khovanov homotopy, strongly invertible knots, and slice disks

October 6, 2024 in the University of Maryland Geometry-Topology Seminar.

Abstract:

Khovanov homology is a refinement of the Jones polynomial of a knot. Recently, there have been a number of exciting applications of Khovanov homology to 4-dimensional topology. In this talk, we will use an indirect approach to re-prove one of these results, that Khovanov homology distinguishes some pairs of disks in the 4-ball. Our proof uses a relationship between the Khovanov homology of a strongly invertible knot and its quotient, coming from a stable homotopy refinement of Khovanov homology.

This is joint work with Sucharit Sarkar. Most of the talk should be fairly broadly accessible; in particular, it will not assume any knowledge of Khovanov homology. Khovanov homotopy

Illustration in Blender

November 8, 2024, an Illustrating Math Seminar Online show-and-ask (5-minute) talk, introducing these tutorials.

Towards a stable homotopy refinement of Legendrian contact homology

November 11, 2024 in the Stony Brook Symplectic / Low-Dimensional Topology Seminar.

Abstract:

The Chekanov-Eliashberg dga, or Legendrian contact homology, was the first modern invariant of Legendrian knots in $\mathbb{R}^3$. This talk is a progress report on a project to give a stable homotopy refinement of Legendrian contact homology, inducing operations like Steenrod squares on linearized Legendrian contact homology. In general, contact homology is defined by counting J-holomorphic curves, but in this case those counts reduce to the Riemann Mapping Theorem and the invariant is described purely combinatorially, from an appropriate knot diagram, and our refinement has a similarly combinatorial flavor. After recalling the basics of Legendrian knot theory and Legendrian contact homology, we will outline our program to refine it, sketch its status, and perhaps describe some examples. This is joint with Lenhard Ng and Sucharit Sarkar. LCH

Khovanov homotopy, strongly invertible knots, and localization

November 14, 2024 in the University of Pennsylvania Geometry-Topology Seminar.

Abstract:

Khovanov homology is a refinement of the Jones polynomial of a knot. Recently, there have been a number of exciting applications of Khovanov homology to 4-dimensional topology. In this talk, we will use an indirect approach to re-prove one of these results, that Khovanov homology distinguishes some pairs of disks in the 4-ball. Our proof uses a relationship between the Khovanov homology of a strongly invertible knot and its quotient, coming from a stable homotopy refinement of Khovanov homology.

This is joint work with Sucharit Sarkar. Most of the talk should be fairly broadly accessible; in particular, it will not assume knowledge of Khovanov homology. Khovanov homotopy

Surfaces in Four-Space

November 20, 2024 in the Rutgers Mathematics Department Colloquium.

Abstract:

The last few years have seen a flury of new results about surfaces in 4-dimensional spaces. In this talk, we will start by discussing various kinds of surfaces and some basic questions about them, like what it means for two of them to be the equivalent. We will then discuss two ways to describe smooth surfaces in 4-space explicitly, give a number of examples illustrating recent results, and mention some still-open questions. The talk will end with a sketch of some of the techniques involved. Most of the results in the talk are due to other people; a few are joint with Sucharit Sarkar.Colloquium / Overview

Even Khovanov homology, odd Khovanov homology

November 20, 2024 in the PATCH Seminar Talk for Graduate Students at Haverford.

Abstract:

I will construct (Khovanov’s) even Khovanov homology and (Ozsváth-Rasmussen-Szabó’s) odd Khovanov homology and talk about some of their similarities and differences. Most of the talk will assume just an understanding of tensor products of abelian groups / vector spaces. Khovanov homology

Local equivalence and Khovanov homology

November 20, 2025 in the PATCH Seminar at Haverford.

Abstract:

The notion of local equivalence has been a key tool in recent work on concordance and homology cobordism groups. In this talk, I will describe a variant of this idea that uses a combination of even and odd Khovanov homology. The result is a group whose elements correspond to equivalence classes of certain simple algebraic structure, and which receives a homomorphism from the smooth concordance group. I will sketch some structures on this group and concrete invariants it defines, and perhaps speculate wildly about other places this strategy might be useful. This is joint work with Nathan Dunfield and Dirk Schütz. Khovanov homology

A local equivalence group from Khovanov homology

January 6, 2025 in the University of Georgia Topology Seminar.

Abstract:

The notion of local equivalence has been a key tool in recent work on concordance and homology cobordism groups. In this talk, I will describe a variant of this idea that uses a combination of even and odd Khovanov homology. The result is a group whose elements correspond to equivalence classes of certain simple algebraic structure, and which receives a homomorphism from the smooth concordance group. I will sketch some structures on this group and concrete invariants it defines, and mention some open questions. This is joint work with Nathan Dunfield and Dirk Schütz. Khovanov homology

Who needs more knot invariants?

January 30, 2025 at the University of Virginia.

Abstract:

Knot theory is the study of smooth, closed loops in 3-space, up to smooth deformation. Much of the subject focuses on knot invariants—algebraic quantities associated to knots. This is often motivated as a way to tell knots apart; but, in practice, we are already extremely good at telling knots apart. So, why do low-dimensional topologists keep looking for new knot invariants?

In the first part of the talk, I will sketch one answer, by describing how knot invariants which are natural, but not too natural, give attacks on some famous problems in smooth 4-dimensional topology. In the second half, I will discuss a couple of recent candidate invariants. The first part draws from work of others; the second is joint with Sarkar and with Dunfield and Schütz.Khovanov homology Khovanov homotopy

The Pairing Theorem and the Fukaya Category

March 26, 2025 at the Princeton Topology Workshop.

Abstract:

We will discuss a Fukaya-categorical interpretation of bordered HF^- with torus boundary and, using it, sketch a proof of the Pairing Theorem. This is joint work with Peter Ozsváth and Dylan Thurston. Bordered Floer