The Southeast Geometry Seminar (SGS) is a semiannual series of one day event sponsored by:
Abstract: With r the natural radial variable on R3, a solution u of the parabolic scalar curvature equation gives a metric g of prescribed scalar curvature on M = Br1 \ Br0 of the form g = u2dr2 + r2γ. When the coordinate spheres are mean convex, which in the case of positive and bounded u is determined entirely by γ, uniform parabolicity is ensured and the existence problem is to take positive data for u at r = r0 and to determine the maximal r1 for which existence holds. If r1 < ∞ then the solution blows up. The purpose of this talk is to discuss a class of metrics for which r1 < ∞, but the blow-up happens in such a way that (M,g) is nonetheless continuously extendible to a manifold with totally geodesic outer boundary at r = r1.
Abstract: It has been established in recent years that there is a range of (high) applied magnetic field values, where superconductivity is confined to a thin sheath near the boundary. The distribution of the superconductivity order parameter over the surface is, however, an open problem in many interesting cases. In the this talk I'll present some recent results with B. Helffer, and show that one may borrow the ideas of superconductivity to obtain similar results on the cholesteric-smectic transition in liquid crystals.
Abstract: For over two centuries, only three examples of complete embedded minimal surfaces of finite topology were known. This talk will describe a surprising new example and how we know it exists.
Abstract: In this talk we will describe the Yamabe invariants and briefly comment on how they have been used to get partial results in the classification of closed 3-manifolds. Then, we will present the construction over manifolds with boundary and show some monotonicity properties with respect to connected sums. As a corollary we show that the invariant of handlebodies is maximal.
Abstract: The Monge-Kantorovich theory of optimal transport has, in recent years, provided a useful framework for analyzing various well-known PDE by reformulating them as gradient flows on certain metric spaces obtained from probability measures. The goal of this seminar is to review some of the geometric structures which underlie this process, then present some new structures which allow for the notion of Hamiltonian flows on these spaces. We will also present some expected applications to PDE. This is joint work in progress with W.Gangbo (GaTech).
Abstract: Inspired by the theory of Diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines in a negatively curved Riemannian manifold. The talk will be a survey on some of our results: We prove a Dirichlet type theorem, define a Hurwitz type constant in terms of the lengths of closed geodesics, and a Khintchine-Sullivan type theorem on the Hausdorff measure of the geodesic lines starting from a cusp that are well approximated by cusp returning ones. This is a joint project with Frederic Paulin (ENS-Paris).