Office Hours: Wednesdays 10-11am and by
appointment
Outline:
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.
Heegaard Floer theory was originally defined using pseudo-holomorphic curves
(solutions to the nonlinear Cauchy-Riemann equations). More recently, a
combinatorial approach to Heegaard Floer theory has been developed, based on
grid diagrams. This shows that the resulting invariants are algorithmically
computable.
This course is meant as an introduction to low-dimensional topology and
Heegaard Floer homology. We will sketch the original definition of the
Heegaard Floer invariants (using symplectic geometry), and then focus on the
combinatorial aspects.
Prerequisites:
Knowledge of manifolds and algebraic topology,
at the level of Math 225.