Titles and Abstracts:
The lectures of Vincent
COLIN (Université de Nantes, IUF) : Contact homology : computations and
applications have been cancelled.
(CNRS, Université de Nantes) : Introduction to Legendrian knots and contact homology
OZSVATH (Columbia University, New York) : An
Introduction to Heegaard-Floer homology
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
Lev ROZANSKY (University of North Carolina,
Chapel Hill) : Matrix factorizations and link
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
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
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,
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.
AUROUX (MIT) : Fukaya categories of symmetric
products and bordered Heegaard-Floer homology
This talk will present an interpretation of the recent work of
Lipshitz-Ozsvath-Thurston extending Heegaard-Floer
3-manifolds with boundary in terms of the symplectic
of symmetric products. More specifically, we will explain
to understand the algebra A(F) associated to a surface in
of the (relative) Fukaya category of the symmetric
JUHASZ (University of
Cambridge, UK) : Sutured Floer homology
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.
Aaron LAUDA (Columbia
University, New York): Categorifying quantum
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
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).
Jacob RASMUSSEN (University of
Cambridge, UK) :
Dehn filling and the Thurston norm
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.
András STIPSICZ (Alfréd Rényi
Institute, Budapest) :
Invariants of Legendrian knots in knot Floer homology
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.