Titles and Abstracts:

Lecture series:

Abstract: Heegaard Floer homology is an invariant defined in a number of low-dimensional situations (three-manifolds, knots, four-manifolds). I will give an introduction to this subject, with special emphasis placed on recent developments. I hope to focus on bordered Heegaard Floer homology, an invariant for three-manifolds with parameterized boundary, which is currently being developed in joint work with Robert Lipshitz and Dylan Thurston.

Abstract: Quantum polynomial invariants of links, such as the Jones polynomial, remain a mysterious phenomenon, since their mathematically rigorous definition is purely combinatorial and their relation to classical topology is unclear. Khovanov's categorification program suggests that these polynomials are graded Euler characteristics of certain graded chain complexes related to links by special combinatorial constructions, and he constructed a chain complex related to the Jones polynomial.

In these lectures I will explain in details how to use simple tools of commutative algebra in order to construct complexes associated to 2-variable and SU(N) HOMFLY-PT polynomials. The categorification of the 2-variable HOMFLY-PT polynomial is based on Soergel's bimodules, while in the SU(N) case we use the so-called matrix factorizations.

I intend to provide the background material on links, braids, quantum polynomials and review basic facts about homological algebra.

Abstract: In this lecture series I will explain from scratch the construction of Khovanov homology (and its different disguises) using representation theory of Lie algebras. We will start by considering the Hecke algebra and its quotient the Temperley-Lieb algebra. I want to explain the notion of higher Schur-Weyl duality and its role in the categorification of link invariants. Finally I will apply this to deduce equivalences of categories between modules for the Lie superalgebra gl(m|n) and a generalised Khovanov algebra.

Additional talks:

Abstract: This talk will present an interpretation of the recent work of Lipshitz-Ozsvath-Thurston extending Heegaard-Floer homology to 3-manifolds with boundary in terms of the symplectic topology of symmetric products. More specifically, we will explain how to understand the algebra A(F) associated to a surface in terms of the (relative) Fukaya category of the symmetric product. Abstract: Sutured manifolds provide powerful tools for studying knots and 3-manifolds. After briefly going through their definition and classical theory, I will define an invariant of sutured manifolds called sutured Floer homology. This is a common extension of the hat version of Heegaard Floer homology and knot Floer homology. It can be used to show that knot Floer homology detects the genus of a knot and also whether the knot is fibered. Furthermore, it helps in the classification of Seifert surfaces that a given knot bounds. Abstract: Crane and Frenkel conjectured that that the quantum enveloping algebra of sl(2) could be categorified at generic q using its canonical basis. In my talk I will describe a realization of this conjecture using a diagrammatic calculus.
If time permits I will also explain joint work with Mikhail Khovanov on how this construction can be generalized to quantum sl(n). Abstract: Suppose Y is a 3-manifold whose boundary is a union of nontrivially linked tori. The Thurston norm on Y measures the geometric complexity (genus) of surfaces representing a given homology class. If Y' is obtained by Dehn filling one boundary component of Y, then the Thurston norm on Y gives an upper bound for the Thurston norm on Y'. I'll explain how sutured Floer homology can be used to show that this bound is an equality for all but finitely many filling slopes. Abstract: An invariant of Legendrian knots will be defined, taking values in the knot Floer homology of the underlying null-homologous knot. With the aid of this invariant we find transversely non-simple knots in many overtwisted contact structures, and show that the Eliashberg-Chekanov twist knots (in particular the 7_2 knot in Rolfsen's table) are not transversely simple.