Low Dimensional Topology

The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, these students will have completed their first year, and in some cases, may already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

The main activity of the Graduate Summer School will be a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

Course Titles and Descriptions:

The 2006 Summer Session in Low Dimensional Topology will consist of seven graduate level lecture series.  On any day during the summer session, three lectures will be offered.  Graduate students are asked to attend the lectures as well as two daily problem sessions associated with the lecture and led by a graduate TA.

John Morgan, Columbia University

Ricci Flow and the Geometrization of three-manifolds

Twenty years ago Thurston conjectured that all three-manifolds can be cut into pieces in a canonical way (decomposing connected sums and cutting open along tori) so that each piece has a (locally) homogenous geometry of finite volume.  This conjecture includes the Poincare Conjecture (that every simply connected three-manifold is diffeomorphic to the three-sphere) as a special case.  We shall cover Perelman's approach to this conjecture.  It goes as follows.  Hamilton introduced a flow equation, call the Ricci flow, for deforming (evolving) a metric on a manifold.  This equation is a 'heat-type' equation and the intuition is that it should smooth out the metric, making it homogeneous.  Hamilton established exactly these sorts of results in various special cases.  The main difficulty in applying the flow in general is is the existence of finite time singularities.  Perelman showed how to do surgery so as to remove the singularities as they form and continue the flow for all time.  Removing the finite time singularities has the effect topologically of doing surgery and also of removing components with homogeneous round metrics.  As tiem converges to infinity one finds tori that cut the manifold into exactly the same pieces conjectured by Thurston.

In this course, we shall examine the basic set-up of the Ricci flow equation and give an overview of Perelman's results.  A basic knowledge of Riemannian geometry (geodesics, the exponential map, Riemannian curvature) as well as the basic facts of three-manifold topology will be assumed.

Mikhail Khovanov, Columbia University

Introduction to Link Homolog

John Etnyre, University of Pennsylvania

Contact geometry in low-dimensions

Recently, contact geometry has been revealing a beautiful and subtle internal structure, and in addition, become a fundamental tool in the study of low dimensional manifolds.  This course will discuss some of the main ideas used in studying contact structures on three-manifolds and in relating contact structures to topology on three-manifolds.  More specifically, after a quick introduction to the subject, we will focus on four areas:

1.  convex surfaces - an important tool for cut and paste type arguments

2.  contact surgery - the contact geometric analogue of Dehn surgery

3.  foliation perturbations - an elegent way to construct contact structures from foliations on a three-manifold

4.  open book decompositions - a topological incarnation of contact structures on a three-manifold

While we will not be able to go into depth on all these topics, we expect to be able to discuss them all to varying levels of detail, and indicate how they are used in some recent advances in contact geometry.

Ron Fintushel, Michigan State University and Ron Stern, University of California Irvine

Six Lectures for Four 4-Manifolds

This lecture series will develop all the known techniques for constructing 4-dimensional manifolds and for altering their underlying smooth structure.  These constructions and their effect are exemplified by particular 4-dimensional manifolds.  Have fun trying to guess which four.

Zoltan Szabo, Princeton University

Lectures on Heegaard Floer Homology

These lectures provide an introduction to Heegaard Floer homology HF(Y) for close oriented three-manifolds Y.  The construction of HF(Y) requires both topological and analytical tools, such as Morse theory, Heegaard diagrams and holomorphic disks.  Additional tools from symplectic geometry (holomorphic triangles, filtrations induced by holomorphic submanifolds) allow extensions of HF(Y) to invariants of knows and links, and in a different direction invarients for smooth cobordisms between three-manifolds.

After studying the background materials, computational tools and various examples, we will look into applications of the Floer homology package.  Some of the applications studied in these lectures will be results of Lens space surgeries, 4-ball genus of knots and unknotting numbers.

David Gabai, Princeton University

Hyperbolic geometry and 3-manifold topology

Starting as the very beginning we will develop the theory of hyperbolic geometry and three-manifold topology needed to prove the Tame Ends Theory (Agol, Calegari- Gabai.)  I.e. "If N is a complete hyperbolic 3-manifold then N is geometrically and topologically tame."  In particular, we will discuss the theory of geometrically finite and infinite hyperbolic 3-manifolds, simplicial hyperbolic sufaces and surface interpolation, end reductions, wild and tame 3-manifolds, and shrinkwrapping.

Cameron Gordon, University of Texas at Austin

Dehn Surgery and 3-Manifolds

Dehn Surgery is the process of removing S^3 a neighborhood of a knot K and gluing it back differently, getting a new 3-manifold M.  Generically, the topological and geometric properties of the complement of K persist in M.  For example, if S^3-K is hyperbolic then usually M is also, and in fact, there seems to be some hope that one can completely determine the exceptions to this general rule.  We will discuss what is known in this direction, and describe some of the techniques that have been used to study this problem.

Participants in the Graduate Summer School also may wish to become involved in the Undergraduate Summer School, attend parts of the Research Program, or participate in the programs of the Education component. Graduate students are expected to participate in Institute-wide activities such as the "Cross Program Activities" and may be asked to contribute some time to volunteer projects related to running the Summer Session.

A limited number of graduate students who have not completed the basic courses may attend. These students will attend some graduate level courses and may be involved as teaching assistants in other programs or work as audio-visual assistants.

The Graduate Summer School is supported by National Science Foundation grant no. 0437137.