No meeting

In this talk, I will introduce the Khovanov-Rozansky knot cohomology theory, and then use it to establish a new bound of the self-linking number of a transversal knot in standard contact $\mathbb{R}^3$, which is sharper than the well known bound given by the HOMFLY polynomial. I will also introduce a sequence of transversal knot invariants, and discuss some of their properties.

This will be an introductory talk about Poisson manifolds, which roughly speaking are manifolds foliated by leaves endowed with a symplectic structures We will see that certain submanifolds, called coisotropic, are interesting from the point of view of reduction. Also, the structure of a Poisson manifold can be encoded in the structure of a Lie algebroid and of the object integrating it, a so-called Lie groupoid. If times permits we will consider the geometry of the sub-algebroids and sub-groupoids associated to coistropic submanifolds, their relation to reduction, and possibly the role played by certain generalizations of coisotropic submanifolds.

Continuation

No meeting

I will briefly explain the Willmore conjecture and then discuss a possible approach using ideas from the theory of spectral curves of Dirac type operators.

Continuation

Continuation

Constructions of Lawson, Kapouleas, and others give examples of surfaces of any genus harmonically mapped into the three sphere. It is not clear from the methods used, however, whether these surfaces occur within any smoothly varying family of harmonic surfaces when the genus is greater than one. I would like to suggest a new method for at least showing existence of (and perhaps constucting) such surfaces which follows Hitchin's proof that the purely algebraic data of a stable Higgs bundle over a Riemann surface give rise to a solution of the dimesionally reduced self-dual SU(2) Yang-Mills equations over the surface. These equations bear a formal resemblence to the equations for a harmonic map of the surface into SU(2) = S^3, and it is hoped that by considering also the data of semistable Higgs bundles, some of the structure of Hitchin's results can be carried over to the harmonic case.

Continuation

In the last fifteen years, 19th century ideas relating integrable systems and conformal geometry have been shown to have discrete analogs. I will relate a well-known algorithm which defines a particular mobius-invariant discrete evolution of a discrete curve on the 2-sphere. I will then indicate a possible generalization to a flow of curves in the 4-sphere imagined as quaternionic projective space. This will involve defining a quaternionic cross ratio and a careful examination of 2-spheres in the 4-sphere via the twistor projection. Time and brain permitting it would be nice to see a glimpse of the "spectral curve" and the associated algebraic integrable system for these flows and observe the relationship with "finite type" solutions for a discrete version of the KdV system.

Continuation of last week's talk.

I'll give definitions, applications and speculations on how to use holomorphic disks to study problems in contact geometry and low-dimensional topology. More details next semester.