SGS XVI: Monday, April 12, 2010
Georgia Institute of Technology

Abstract: Asymptotically flat manifolds are models of isolated systems in general relativity. We will discuss the foliation by stable spheres with constant mean curvature in asymptotically flat manifolds. We will also talk about the physical motivation and discuss how the foliation relates to the concept of center of mass in general relativity.

Abstract: We will discuss recent results which provide rigorous restrictions on the geometry of static and stationary n-body solutions in relativity for various matter fields. Much of the talk will describe joint work with Robert Beig and Gary Gibbons.

Abstract: Almost-convex subsets of a metric space are defined in terms of intrinsic distance functions. I will discuss how they arise when considering limits of smooth Riemannian submanifolds and some of their applications.

Abstract: I will talk about joint with R. Schoen on a spectral problem for manifolds with nonempty boundary. We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces with boundary we obtain an upper bound on the first Steklov eigenvalue in terms of the genus and the number of boundary components of the surface. This generalizes a result of Weinstock from 1954 for surfaces homeomorphic to the disk. We attempt to find the best constant in this inequality for annular surfaces. Motivated by the annulus case, we explore an interesting connection between the Dirichlet-to-Neumann map and minimal submanifolds of the ball that are solutions to the free boundary problem. We then prove general upper bounds for the first Steklov eigenvalue for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities.

Abstract: For oriented knots, the question of whether a knot is invertible (isotopic to itself with the opposite orientation) and/or chiral (not isotopic to its mirror image) have great implications in topology and in other sciences. We consider the case of oriented links and ask similar questions: if we invert one or more components, is this isotopic to our original link? What if we permute components? Or take a mirror image? The group of transformations of this type which can be realized by an isotopy of the link is called the intrinsic symmetry group of that link. We present the first computations of the intrinsic symmetry groups of links with 8 and fewer crossings.

Our motivation in understanding these symmetries is to calculate a geometric property of knots: the ropelength of a knot is the minimum amount of rope (of fixed radius 1) needed to construct that knot.

The traditional definition of the symmetry group of a link is the mapping class group MCG(S³, L) of the pair S³, L. Our symmetry groups are the images of the traditional symmetry groups of links under the natural homomorphism from MCG(S³, L) onto MCG(S³) X MCG(L).

This is joint work with Jason Cantarella and many students.

Abstract: In 1854 Riemann extended Gauss' ideas on curved geometries from two dimensional surfaces to higher dimensions. Since that time mathematicians have tried to understand the structure of geometric spaces based on their curvature properties. It turns out that basic questions remain unanswered in this direction. In this lecture we will give a history of such questions for spaces with positive curvature, and describe the progress that has been made as well as some outstanding conjectures which remain to be settled.

Click here for the poster for this event.