Matt Hedden, Heegaard-Floer homology
An important philosophy which has arisen during the past 20 years is the idea that one should
use tools from symplectic geometry to address questions in low-dimensional topology
(which at first glance often appear to have nothing to do with symplectic geometry).
The most dramatic instance of this philosophy occurs in the form of Ozsvath and
Szabo's Heegaard Floer homology, which uses a manifestly symplectic construction
to produce an extremely powerful package of invariants for low-dimensional topology.
These invariants have led to a dazzling array of applications, and have had important
implications for the study of Dehn surgery, knots and links, contact structures, and
smooth 4-manifolds and the surfaces embedded therein.
This mini-course will introduce participants to the Heegaard Floer homology invariants. No prior knowledge of these invariants will be assumed, and the course will be accessible to fresh Ph.D. students. The course will give an overview of what these invariants offer to a low-dimensional topologist, and will then introduce and develop their denitions and basic properties. Special attention will be paid to the knot and link invariants coming from this theory, and their applications. A guiding theme which will be developed throughout the course is that Heegaard Floer homology carries a great deal of information about homologically essential surfaces, and that this information lies at the heart of many of the theory's striking applications.
Jacob Rasmussen, Khovanov homology and its offspring
I intend to start simply, with the definitions of the Jones polynomial and Khovanov
homology, and then spend some time discussing the Witten-Reshitikhin-Turaev philosophy for polynomial invariants of knots in the 3-sphere and the general project of
categorifying it. Next, I will discuss some aspects of this program which we understand rather well - namely, categorifying the polynomials associated to the standard
representation of sl(n), and to the standard HOMFLY polynomial. The course will
end by discussing some aspects of the theory of colored homologies, which are not so
well understood and offer many interesting prospects for future research.
A theme which I hope to emphasize in the course is that we should aim to do more than simply define new invariants of knots; we should hope to understand them. In the study of quantum invariants, the general pattern has been that definitions have gotten far ahead of understanding. Categorification provides an additional level of structure which yields both geometric applications (e.g. bounds on the slice genus, unknot detection), and insight into the behavior and meaning of the original polynomial invariants.
Chris Wendl, Contact 3-manifolds and holomorphic curves
This minicourse is intended essentially as a `user's guide' to the intersection theory of punctured holomorphic curves and its applications in 3-dimensional topology.
Intersection theory has played a prominent role in the study of closed symplectic
4-manifolds since Gromov's 1985 paper on J-holomorphic curves, leading to a myriad of beautiful rigidity results that are either not accessible or not true in higher
dimensions. In recent years, Siefring's highly nontrivial extension of this theory to the
punctured case has led to similarly beautiful results about contact 3-manifolds and
their symplectic fillings.
I will begin with a brief summary of the closed case and an easy application (McDuff's characterisation of symplectic ruled surfaces), and then explain the essentials of Siefring's intersection theory and how to use it in the real world. As a sample application, I will discuss the classification of symplectic fillings of planar contact manifolds via Lefschetz fibrations.
Prior knowledge of punctured curves or the analytical subtleties of the theory is not required; I will state the required definitions and take most of the hard analytical results as black boxes in order to focus instead on topological issues.