Robert Lipshitz's 2015-2016 Tour

Dates and venues

Abstracts

August 31, Berkeley Symplectic Geometry Seminar.
Hochschild homology and Smith theory
Abstract. We will explain a localization result for Z/2-equivariant Hochschild homology, some applications of the result to Heegaard Floer homology, and some speculation about how it could be applied more broadly using quilted Floer homology. The title notwithstanding, both the main results and the talk will be quite concrete. This is joint work with David Treumann.

September 2, Berkeley RTG Seminar.
An introduction to Heegaard Floer homology
Abstract. We will sketch some of the formal structure and information content of Heegaard Floer homology, and end with a hint at its definition. The talk will assume a knowledge of singular homology and the definition of smooth manifolds.

September 2, Berkeley Topology Seminar.
A concordance homomorphism from equivariant Heegaard Floer homology
Abstract. We will define yet another smooth concordance homomorphism from Heegaard Floer homology, using yet another definition of Z/2-equivariant Floer homology. This is joint work with Kristen Hendricks and Sucharit Sarkar.

October 13, University of Oregon Topology Seminar.
Periodic knots and Hochschild homology
Abstract. We will start by reviewing the definition of a 2-periodic knot in S³, and two restrictions on periodic knots: Murasugi’s condition, in terms of the Alexander polynomial, and Edmonds’s condition, originally proved via minimal surfaces. We will then explain a common generalization, in terms of Heegaard Floer homology, due to Hendricks. Hendricks’s proof is, at heart, analytic. We will end with a sketch of how one can recover cases of Hendricks’s result, and similar results, using results on Hochschild homology and the formal structure of bordered Heegaard Floer homology. The last part of the talk is joint work with David Treumann.

November 6, Cascade Topology Seminar (Portland State).
Properties of the Alexander polynomial and knot Floer homology
Abstract. We will start by recalling a definition of the Alexander polynomial and Manolescu-Ozsváth-Sarkar’s combinatorial definition of its categorification, knot Floer homology. We will then discuss some geometric properties of the Alexander polynomial which have been lifted to knot Floer homology, and some applications of these, and some properties which have not yet been lifted.

December 1, University of Oregon Geometric Analysis Seminar.
Holomorphic curves, Floer theory, and degenerations
Abstract. Lagrangian intersection Floer homology is an invariant of Lagrangian submanifolds defined by counting pseudo-holomorphic curves (solutions to a particular elliptic PDE). It can be used to define invariants of 3-manifolds in several ways; one such invariant is called Heegaard Floer homology. In this talk, we will recall the basic results about moduli spaces of pseudo-holomorphic curves needed to define Lagrangian intersection Floer homology. We will then talk about a second framework for these results, and the underlying analysis needed to define an extension of Heegaard Floer homology, called bordered Heegaard Floer homology. (This is joint work with Peter Ozsváth and Dylan Thurston.) Finally, we will mention some (slightly vague) open problems which seem to require further analytic developments.

January 26, University of Oregon Geometry / Topology Seminar.
Introduction to (Heegaard) Floer homology
Abstract. Roughly, Lagrangian intersection Floer homology associates to a pair of Lagrangian submanifolds L, L' a graded vector space HF(L,L'), whose Euler characteristic is the intersection number of L and L'. Lagrangian intersection Floer homology can be used to define invariants of knots, 3-manifolds, and 4-manifolds, in several ways; one is called Heegaard Floer homology. Some of these invariants now have elegant combinatorial definitions. In this talk, we will discuss the basic definitions of Lagrangian intersection Floer homology and illustrate how the play out in a combinatorial definition of the Heegaard Floer knot invariant. This may be the first talk in a series of two.

February 4, Boston College Geometry / Topology Seminar.
A flexible construction of equivariant (Heegaard) Floer homology.
Abstract. For G a finite group we will sketch a construction of G-equivariant Lagrangian Floer homology which enjoys good invariance properties. Specializing to the case of Heegaard Floer homology we will indicate how this construction leads to some apparently new concordance invariants. This is joint work with Kristen Hendricks and Sucharit Sarkar.

February 11, University of Texas at Austin Geometry Seminar.
A flexible construction of equivariant Floer homology, and applications.
Abstract. For G a finite group acting on a symplectic manifold M and preserving a pair of Lagrangian submanifolds L, L' we will outline a new construction of the G-equivariant Floer homology of L and L', with good invariance properties, and mention some applications of Heegaard Floer homology. This is joint work with Kristen Hendricks and Sucharit Sarkar.

February 23, University of Oregon Geometry / Topology Seminar.
Introduction to (Heegaard) Floer homology, part 2.
Abstract. We will continue where we left off on January 26, by sketching the proofs that d²=0 on the Lagrangian intersection Floer complex and that Lagrangian intersection Floer homology is independent of the choice of almost complex structure. We will then discuss aspects of the theory which are determined by the algebraic topology of the path space, namely, the grading and when the theory can be defined over Z (instead of Z/2). Time permitting, we will briefly introduce the Fukaya category and a sense in which the self-Floer homology of a Lagrangian is a deformation of singular (co)homology, and perhaps mention homological mirror symmetry.

April 2, Graduate Student Topology Conference (Indiana University).
The Alexander polynomial and knot Floer homology.
Abstract. Knot Floer homology is a re nement of the Alexander polynomial of a knot, introduced around 2004 by Ozsváth-Szabó and Rasmussen. In this talk, we will discuss some properties of the Alexander polynomial which carry over to knot Floer homology, and some properties which are not (yet) known to carry over. In the process, we will see a variety of classical and modern applications of the two invariants. Most of the results are due to others, but a few are joint with Treumann and Hendricks-Sarkar.

July 30 -- August 2, Homological Invariants in Low Dimensional Topology (IPM, Tehran, Iran).
Bordered Heegaard Floer homology.
Abstract. After a short introduction to bordered Floer homology I will talk about using it to compute HF^ and then extending it to HF^- in the case of torus boundary. This is joint work with Peter Ozsváth and Dylan Thurston.

Previous years: 2015-2016.