March 26, 2025
Lecture 1: From Heegaard diagrams to holomorphic disks
Friday, May 9th, 4:00 PM, Eckhart 206, 5734 S. University Ave
Heegaard Floer homology is a tool for studying three- and four-dimensional manifolds, using methods that are inspired by symplectic geometry. There are further manifestations of this theory, giving invariants for knots in three-manifolds (knot Floer homology), and three-manifolds with parameterized boundary (bordered Floer homology). I will describe both topological applications of this theory and certain computational advances which relate Floer homology with other algebraic objects appearing in symplectic geometry. Heegaard Floer homology was discovered in joint work with Zoltan Szabo.
Lecture 2: A knot invariant from grid diagrams
Monday, May 12, 4:00 PM, Ryerson 251, 1100 E. 58th St.
Knot Floer homology is an invariant for knots in three-space, defined by a suitable adaptation of Heegaard Floer homology. It has the form of a bigraded vector space, encoding information about the complexity of the knot, an enrichment (or "categorification") of the Alexander polynomial. The invariant was originally defined in collaboration with Zoltan Szabo, and indepedently by Jacob Rasmussen. I will describe a purely combinatorial formulation of this invariant, discovered in joint work with Ciprian Manolescu and Sucharit Sarkar, and further elaborated in joint work with Manolescu, Szabo, and Dylan Thurston. I will also sketch some of the applications of this invariant to knot theory.
Lecture 3: Bordered Floer homology
Tuesday, May 13, 4:00 PM, Eckhart 202, 5734 S. University Ave
Bordered Floer homology is an invariant for three-manifolds, which sets up a Mayer-Vietoris like description of Heegaard Floer homology. To a surface \(S\), this theory associates an algebra \(A(S)\); to a bordered three-manifold with boundary \(S\), it associates a module over \(A(S)\); and to a pair of three manifolds, each with boundary \(S\), it expresses the Heegaard Floer homology of the glued up closed three manifold in terms of the modules associated to the two pieces. This invariant was formulated and computed in several cases, including, originally, for the \(U=0\) specialization of Heegaard Floer homology. I hope to describe some recent developments for the unspecialized theory. This is joint work with Robert Lipshitz and Dylan Thurston.
Peter Ozsvath (Princeton University)
Exact times and locations to be announced.