Office Hours: Fridays 2 - 3:30 pm and by
appointment
Heegaard Floer theory, developed by Ozsvath and Szabo, is a powerful
technique for studying the key objects in low-dimensional topology: knots
and links in the three-sphere, 3- and 4-dimensional manifolds.
In particular, it provides answers to the following questions: Given a
knot in space, how can we tell if it is the unknot? Given a
two-dimensional homology class in a three-manifold, what is the minimal
complexity of a surface representing that class? How can one distinguish
4-manifolds that are homeomorphic but not diffeomorphic?
While the questions above can also be answered in different ways, Heegaard
Floer theory provides a unified approach to them, as well as to many other
problems.
This course is meant as an introduction to low-dimensional topology and
Heegaard Floer homology. We will explain both the original definition of
the Heegaard Floer invariants (using symplectic geometry), and some of the
more recent combinatorial definitions.
Prerequisites: Math 225A (Differentiable Manifolds), Math 225C
(Introduction to Algebraic Topology).