Can the symplectic structure of a cotangent bundle recover the smooth structure of the underlying manifold? This is a motivating question for this topic. On the differentiable topology side, D. Sullivan and others have developed string topology. On the symplectic geometry side, Y. Eliashberg and H. Hofer have developed symplectic field theory. I will discuss joint work with D. Sullivan trying to connect the two theories. In the first talk, I will introduce an under-construction extension of (open) string topology. In the second talk, I will introduce Legendrian contact homology (including all relevant definitions from sympletic geometry) and try to relate the two theories.

Continuation.

With the exception of spheres, unduloids are the simplest examples of constant mean curvature (CMC) surfaces and the only other CMC surfaces of revolution. Kapouleas showed that unduloids can be cut and pasted to form more complex CMC surfaces and in this way, comprise the basic building blocks for a significant class of CMC surfaces. In order to get a handle on the geometry and topology of moduli spaces of such "cut-and-paste" surfaces, we study the basic building blocks of these surfaces, the unduloids themselves. To simplify the problem further, we focus on the unduloids that share a fixed axis of revolution.

Define $\mathcal{U}_{H}$ to be those unduloids which have mean curvature $H$ and the $z$-axis as their axis of revolution. Then $\mathcal{U}_{H}$ is naturally parametrized by the unit disk and this parametrization gives rise to the (not quite global) polar coordinate system $\left(e,\theta\right)$. Furthermore, $\mathcal{U}_{H}$ possesses a canonical symplectic form which has a fairly simple coordinate expression relative to $\left(e, \theta\right)$. This form can be integrated over $\mathcal{U}_{H}$ to show that the total area of $\mathcal{U}_{H}$ is finite.

Continuation

We start by introducing the Hofer norm (bi-invariant) on the Hamiltonian diffeomorphism group $Ham(M,\omega)$. We then study the symplectomorphism group $Symp(M,\omega)$ in the framework of Hofer's geometry. In particular, we will consider the bounded isometry conjecture of Lalonde and Polterovich. A related question of extending the Hofer norm to $Symp(M,\omega)$ will also be addressed. If time permits, we will consider other bi-invariant norms on $Symp(M,\omega)$ and discuss some open questions.

Continuation.

My talk is concerned with the study of finite group actions on 4-manifolds. The problem of group actions on manifolds dates back at least to the 19th century and was strongly influenced by Hilbert's 5th problem. By the first half of the 20th century it has developed into a well recognized discipline, which has connections with many other areas of mathematics. On the other hand, one only begins to understand the topology of 4-manifolds in a more systematic way after work of Freedman and Donaldson in the beginning of 1980's. In the past decade, more attention has been given to the topology of symplectic 4-manifolds. I will review some of the basic questions, ideas and tools available in both subjects (group actions and topology of 4-manifolds), and survey some of the major results about finite group actions on 4-manifolds obtained in the past two decades.

Continuation.

Beginning with a reprise of a talk from last semester, I'll breifly review a gauge theoretic model for harmonic maps from a surface into S^3, where harmonicity is correlated to the flatness of a certain loop of connections on a complex vector bundle over the surface. The flatness of such a loop allows it to be transformed into a another loop which describes the complex structure and "Higgs field" of a bundle which is semistable, in a sense introduced by Hitchin in his work on the seld-dual Yang-Mills equations over a surface. The hope is to be able to reverse this process using the non-Abelian Hodge theory developed by Simpson and others. This could provide means for constructing higher genus harmonic surfaces in S^3 and parametrizing their moduli.

Gobble-gobble, gobble-gobble.

In this talk, I will prove the invariance of perturbations of the Khovanov-Rozansky cohomology, and use it to generalize the Rasmussen invariants. I will also discuss its application in contact topology.

I will discuss joint work with R. Kusner on an alternative geometric construction to thick knots (which we term "raceways") and focus on the formulation of raceways and demonstrating an asymptotic upper bound for crossing number relative to raceway length. With luck, there may also be a discussion of similar scope about Kauffman's related "ribbons" construction.

Continuation.