TBA

Enjoy the snow and Valentine's day.

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 constructing) 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 dimensionally reduced self-dual SU(2) Yang-Mills equations over the surface. These equations bear a formal resemblance 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.

TBA

Abstract: We study the rigidity of complete embedded constant mean curvature surfaces in space. One of the results that we prove is the following: Let M be a complete embedded constant mean curvature surface of finite genus. If M has bounded norm of the second fundamental form and M is not the helicoid then M is rigid. Here rigid means that the inclusion map of M into space represents the unique isometric immersion of M into space with the same constant mean curvature up to ambient isometries. A key ingredient in the proof is our recent Dynamics Theorem for CMC surfaces which we are going to discuss during the talk. This is joint work with Bill Meeks.

Enjoy your spring break.

Abstract: One of the basic questions in the theory of differentiable transformation groups is how the groups may depend on the underlying differentiable structure. Such questions have been extensively studied in the past, mainly focusing on the case of homotopy spheres of dimension greater than or equal to 7. On the other hand, differentiable transformation groups of exotic 4-manifolds have remained to be a rather untest territory. In this talk, we will discuss some recent progress in this direction. This is joint work with Slawomir Kwasik.

Continuation.

Abstract: Some generalizations and variations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are cyclic, then these surfaces are topologically unknotted. Using a twist-spinning construction from high-dimensional knot theory, we construct examples of knotted surfaces whose complements have cyclic fundamental groups.

Abstract: If one searches for k-dimensional submanifolds with critical k-dimensional volume in a Riemannian manifold, then one is led towards elliptic partial differential equations involving the mean curvature vector of the submanifold. I will present new constructions of volume-critical submanifolds of the sphere in two contexts: hypersurfaces with constant mean curvature in spheres of any dimension; and Legendrian submanifolds in spheres of odd dimension that are stationary under variations preserving the contact structure. These are constructed by solving the associated elliptic PDE using singular perturbation theory. I will then highlight some of the analytic and geometric similarities between these two contexts.

Abstract: I'll show how to construct many simply connected and non-simply connected symplectic and smooth manifolds, including a minimal symplectic manifold homeomorphic to CP^2#3(-CP^2) containing a symplectic genus 2 surface with simply connected complement. This is joint work with Paul Kirk.

TBA

The existence or non-existence of Einstein metrics on a topological 4-manifold is strongly related to the differential structure considered. We show that there exist infinitely many topological 4-manifolds such that each manifold admits a smooth structure which supports an Einstein metric and has infinitely many other structures on which no Einstein metric can exist. We complete this result with theorems about non-existence of Einstein metrics on manifolds with finitely presented fundamental group and discuss some nice corollaries. The main tools are Seiberg-Witten theory, cyclic coverings of complex surfaces and symplectic surgeries.