Southeast Geometry Seminar

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

The Southeast Geometry Seminar (SGS) is a semiannual series of one day event sponsored by:

The organizers are: Vladimir Oliker (Emory), Mohammad Ghomi and John McCuan (Georgia Tech), Fernando Schwartz (UTK), and Gilbert Weinstein (UAB).

We have two recent features:

  1. A poster session. Please contact us if you would like to present a poster.
  2. An afternoon public lecture by our plenary speaker, Richard Schoen. Click here for the poster for this event.

If you plan to participate, please help us prepare for the event by registering.

Thanks to an NSF grant, we have funds to support participants, particularly students and recent Ph.D. recipients. We encourage women and minorities to apply. To apply please write to us, or fill out the appropriate section on the registration form.

8:00 AM - Poster Session & Refreshments - Skiles 269

8:30 AM - Morning Session - Skiles 269

8:30 AM - Lan-Hsuan Huang (Columbia U)
Constant mean curvature foliations for isolated systems

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.

9:30 AM - Richard Schoen (Stanford U)
On the geometry of static and stationary spacetimes

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.

10:30 AM - Jeremy Wong (U of Georgia)
Applications of Almost-Convexity.

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.

12:00 PM - Poster Session - Skiles 269

12:00 PM - Lunch

2:00 PM - Afternoon Session - Skiles 269

2:00 PM - Aliana Fraser (U of British Columbia)
The first eigenvalue of the Dirichlet-to-Neumann map and minimal surfaces

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.

3:00 PM - Jason Parsley (Wake Forest U)
Intrinsic symmetries of knots and links

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.

5:00 PM - Evening Session - 1116W Klaus Building

5:00 PM - Public Lecture: Richard Schoen (Stanford U)
Spaces of positive curvature

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.

Authored by: Gilbert Weinstein
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