*The 2010 Chern Lectures will be delivered by Peter S. Ozsvath on April 13, 15, 20 and 22, 2010.*

Department of Mathematics, University of California, Berkeley, presents

The 2010 Chern Lectures

Peter S. Ozsvath

Columbia University**Heegaard Floer homology****Lecture 1:** Introduction to Heegaard Floer homology

Tuesday, April 13, 2010

4:00 - 5:00 pm

Sibley Auditorium, Bechtel Hall

Heegaard Floer homology is an invariant for low-dimensional manifolds defined using methods from symplectic geometry (holomorphic disks, Lagrangian Floer homology). To a closed, oriented three-manifold, this invariant associates a module over the polynomial algebra in a formal variable U. I will outline the structure of this theory and discuss various of its topological applications. This construction (as an invariant for three- and four-manifolds) was originally discovered in collaboration with Zoltán Szabó. The generalization to knots was discovered independently by Jacob Rasmussen.**Lecture 2:** Knot Floer homology

Thursday, April 15, 2010

4:00 - 5:00 pm

Sibley Auditorium, Bechtel Hall

Heegaard Floer homology can be used to define an invariant for knots in the three-sphere. This invariant, whose definition involves holomorphic curves, admits an elementary, combinatorial description in terms of grid diagrams. I will describe the construction, and sketch a combinatorial proof of its topological invariance. This lecture will cover joint work with Ciprian Manolescu, Sucharit Sarkar, Zoltán Szabó, and Dylan Thurston.**Lecture 3:** Bordered Floer homology

Tuesday, April 20, 2010

4:00 -5:00 pm

Sibley Auditorium, Bechtel Hall

I will describe an extension of the U=0 specialization of Heegaard Floer homology, **HF**, to three-manifolds with parameterized boundary. The resulting theory, *bordered Floer homology*, associates a differential algebra to an oriented two-manifold, and a differential module over that algebra to a three-manifold with boundary. These differential modules can also be used to reconstruct **HF** for closed three-manifolds. This is joint work with Robert Lipshitz and Dylan Thurston.**Lecture 4:** Heegaard Floer homology and surgery formulas

Thursday, April 22, 2010

4:00 - 5:00 pm

Room 60, Evans Hall

After setting up some necessary background, I will turn attention to surgery formulas for reconstructing the Heegaard Floer homology groups of three-manifolds obtained as surgeries on knots and links. I will then explain how this can be used to, in principle, compute Heegaard Floer homology groups of closed (three- and four-) manifolds from data associated to grid diagrams.