Talks :: Robert Lipshitz

Real bordered Floer homology

September 9, 2026 in the Boston College Geometry/Topology Seminar.

Abstract:

Real Seiberg-Witten theory and real Heegaard Floer homology are relatively new ways of integrating $\mathbb{Z}/2$ symmetries into these powerful invariants, and real Seiberg-Witten invariants, in particular, have already had dramatic applications. In this talk, I will briefly survey different ways $\mathbb{Z}/2$ symmetries can be reflected in Heegaard Floer theory, and then discuss joint work with Peter Ozsváth on extending real Heegaard Floer homology to 3-dimensional cobordisms. Bordered Floer

Real bordered Floer homology

October 13, 2026 in the Stanford Topology Seminar.

Abstract:

After very briefly recalling Guth-Manolescu’s real Heegaard Floer homology, we will discuss an extension of the “hat” version of it to 3-manifolds with boundary and discuss what we know about using it for computations. This is joint work with Peter Ozsváth. Bordered Floer

There’s plenty more where those came from

October 20, 2026 in the Michigan State Topology Seminar.

Abstract:

I will construct at least 6 new concordance homomorphisms coming from Khovanov homology. This is an extension of work with Dunfield-Schütz, building on ideas of Dai-Hom-Stoffregen-Truong and others, with most research conducted by Anthropic’s Claude LLM. At the end, I will mention some open problems, briefly mention a few thoughts about the ethics of this project, and convey some optimism that mathematics is, in fact, far from finished. Khovanov homology

There’s plenty more where those came from

October 26, 2026 in the Oregon State Geometry/Topology Seminar.

Abstract:

I will construct at least 6 new concordance homomorphisms coming from Khovanov homology. This is an extension of work with Dunfield-Schütz, building on ideas of Dai-Hom-Stoffregen-Truong and others, with most research conducted by Anthropic’s Claude LLM. At the end, I will mention some open problems, briefly mention a few thoughts about the ethics of this project, and convey some optimism that mathematics is, in fact, far from finished. Khovanov homology

There’s plenty more where those came from

October 27, 2026 in the University of Oregon Topology Seminar.

Abstract:

I will construct at least 6 new concordance homomorphisms coming from Khovanov homology. This is an extension of work with Dunfield-Schütz, building on ideas of Dai-Hom-Stoffregen-Truong and others, with most research conducted by Anthropic’s Claude LLM. At the end, I will mention some open problems, briefly mention a few thoughts about the ethics of this project, and convey some optimism that mathematics is, in fact, far from finished. Khovanov homology

Surfaces and the Fourth Dimension

September 8 – November 24, 2025, a series of eight lectures as the Minerva Visitor at Princeton University. Lecture plan and exercises.

Concordance, Khovanov homology, and local equivalence

October 17, 2025 in the New Mexico State Geometry/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 recall the basics of Khovanov homology, and then give a simple example of a local equivalence group constructed from it. 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. The talk will conclude with a discussion of how effective this invariant is for studying knot cobordism, and maybe some little open questions. This is joint work with Nathan Dunfield and Dirk Schütz. Khovanov homology

Real bordered Floer homology

January 12, 2026 in the Duke Geometry/Topology Seminar.

Abstract:

Real Heegaard Floer homology is a new invariant of branched double covers, introduced by Gary Guth and Ciprian Manolescu, and inspired by work of Jiakai Li and others in Seiberg-Witten theory. After sketching their construction, we will describe an extension of the “hat” variant to 3-manifolds with boundary, and the algorithm this gives to compute it when the fixed set is connected. We will end with some open questions. This is joint work with Peter Ozsváth. Bordered Floer

How Surfaces Can Be Exotic

January 15, 2026 in the Dartmouth Mathematics Department Colloquium.

Abstract:

Smooth surfaces in 4-space are a mathematician’s wonderland: simple to describe explicitly, but exhibiting a baffling range of unintuitive behavior. The last five years have seen remarkable advances in our understanding, but still many basic questions remain, for now, out of reach. In this talk, I will try to convince you that one really can get one’s mind, if not one’s hands, on surfaces in 4-space; that we know a lot of counterintuitive properties they have; and that there is a huge amount more to explore. A beginning course in multivariable calculus (up to the second derivative test) will be helpful in following the talk, but most of it will not assume background beyond that.Colloquium / Overview

Real bordered Floer homology

April 13, 2026 in the Georgia Tech Geometry Topology Seminar.

Abstract:

Real Heegaard Floer homology is a new invariant of branched double covers, introduced by Gary Guth and Ciprian Manolescu, and inspired by work of Jiakai Li and others in Seiberg-Witten theory. After sketching their construction, we will describe an extension of the “hat” variant to 3-manifolds with boundary, and the algorithm this gives to compute it when the fixed set is connected. We will end with some open questions. This is joint work with Peter Ozsváth. Bordered Floer

Symmetries and Heegaard Floer homology

May 5, 2026 in the CUNY Geometry/Topology Seminar.

Abstract:

Many of the recent developments in Floer-theoretic invariants of 3- and 4-manifolds have used symmetries under finite group actions – especially $\mathbb{Z}/2$ . We will start by surveying some of those kinds of symmetries and applications. Then, we will talk about three different ways of encoding $\mathbb{Z}/2$ symmetries in Heegaard diagrams, briefly introduce Guth-Manolescu’s real Heegaard Floer homology, and sketch an extension of it to 3-manifolds with boundary. The last part is joint work with Peter Ozsváth; there is related work by Binns-Guth-Xiao. Bordered Floer

Towards Legendrian contact homotopy

May 21, 2026 at the Georgia Topology Conference.

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 homology and surfaces

July 13 – 17, 2026, five lectures at the Park City Mathematics Institute Graduate Program.

Abstract:

Khovanov homology of surfaces in 4-space has recently become an important tool, both because of its direct applications to distinguishing smooth surfaces and the indirect role it plays in the construction of the skein lasagna invariants of 4-manifolds. In this series, we will recall the construction of Khovanov homology for knots in 3-space and surfaces in 4-space, use the latter to distinguish some exotic surfaces, and survey the definition and known examples and applications of the skein lasagna module of a 4-manifold. Khovanov homology

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 7, 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

December 6, 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

December 6, 2024 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 in the University of Virginia Mathematics Department Special Colloquium.

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

Talks

Talks AY 2026-2027

Talks AY 2025-2026

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