Course Abstracts

Instructor: Zoltán Szabó
Title: Introduction to Heegaard Floer homology

The aim of this course is to give a quick introduction to Heegaard Floer homology for closed, oriented three-manifolds. Topics will include: Heegaard diagrams, symplectic manifolds, Lagrangian Floer homology, variants of Heegaard Floer homology, topological invariance, examples and computations.

Suggested reading:

P. Ozsváth and Z. Szabó, Heegaard diagrams and holomorphic disks, Arxiv, math.GT/0403029


Instructor: Peter Ozsváth
Title: Heegaard Floer homology II

This course is a continuation of the "Introduction to Heegaard Floer homology". I will describe further topics in the theory. The topics I hope to cover include: surgery long exact sequences, absolute gradings, invariants for knots, and the invariants for contact structures.

A survey paper can be found on the net:

P. Ozsváth and Z. Szabó, Heegaard diagrams and holomorphic disks, Arxiv, math.GT/0403029.

Other relevant papers include:

P. Ozsváth and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math. 173 (2003), no. 2, 179 — 261.

_____, Heegaard Floer homology and alternating knots. Geom. Topol. 7 (2003), 225 — 254.

_____, Holomorphic disks and knot invariants, math.GT/0209056.

J. Rasmussen, Floer homology and knot complements, Harvard thesis, math.GT/0306378.

Instructor: Cameron Gordon
Title: 3-Manifolds and Dehn surgery

We will describe the conjectured picture of 3-manifolds, and discuss the Dehn filling construction. A particular focus will be the Dehn surgeries on hyperbolic knots in the 3-sphere that give non-hyperbolic 3-manifolds. We will also describe some of the combinatorial surface-intersection techniques that have been used to study Dehn filling.

A couple of general references are:

C. McA. Gordon, Dehn filling: a survey, in Knot Theory, Banach Center Publications Vol 42, 1998, 129-144.

_____, Combinatorial methods in Dehn surgery, Lectures at Knots '96, Series on Knots and Everything Vol 15, 263-290.

Instructor: Ronald Fintushel
Title: Knot Surgery Revisited

These lectures will be an introduction to the use of Seberg-Witten invariants to study smooth 4-manifolds. To do this, we will study the calculation of the Seiberg-Witten invariant of the result of knot surgery and its relation to the Alexander polynomial from several different points of view: 4-dimensional S-W theory, a 3-dimensional approach related to S^1-valued Morse theory, and via Taubes' SW = Gromov theorem. We will also point out some applications.

Background: No previous acquaintance with Seiberg-Witten theory will be assumed, but, of course, it would be helpful to already know something about these equations. A good source is John Morgan's book:

""The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds", Princeton Univ Pr. or the survey paper:

S. K. Donaldson, 'The Seiberg-Witten equations and 4-manifold topology', Bull. Amer. Math. Soc. 33 (1996), 45-70.

Relevant papers are:

R. Fintushel and R. Stern, 'Knots, links, and 4-manifolds', Invent. Math.134 (1998), 363-400.

G.Meng and C. Taubes, 'SW = Milnor torsion', Math. Res. Lett. (1996), 661-674.

S. K. Donaldson, 'Topological field theories and formulae of Casson and Meng-Taubes', in "Proceedings of the Kirbyfest" (Berkeley, CA, 1998)," 87 — 102, Geom. Topol. Monogr., 2.

C. Taubes,'The geometry of the Seiberg-Witten invariants', in "Surveys in differential geometry, Vol. III" (Cambridge, MA, 1996), 299-339, Int. Press.

Program