Program
(All events, including breakfast, lunch and snacks, in Ayres 405)

7:30-8:30am
Posters, Registration and Breakfast

8:30-9:30am
Davi Maximo: "On the blow-up of four-dimensional Ricci flow singularities."

9:45-10:45am
Ailana Fraser: "Eigenvalue problems and minimal surfaces"

10:45-11:15am
Coffee Break

11:15am-12:15pm
John Pardon: "Totally disconnected groups (not) acting on three-manifolds."

12:30-1:45pm
Lunch

1:45-2:45pm
Christine Breiner: "Gluing constructions for embedded constant mean curvature surfaces."

3-4pm
Marcelo Disconzi: "On the Einstein equations for relativistic fluids."

4-6pm
Snacks




Abstracts

Christine Breiner, Columbia

"Gluing constructions for embedded constant mean curvature surfaces."

Abstract: Constant mean curvature (CMC) surfaces are critical points to the area functional with an enclosed volume constraint. Classic examples include the round sphere and a one parameter family of rotationally invariant surfaces discovered by Delaunay. In this talk I outline the gluing method we develop that produces a large class of new examples of embedded CMC surfaces of finite topology. We refine the method first developed by Kapouleas in 1990, that produced immersed examples, using fundamental ideas he later introduced to construct a larger class of immersed examples. I will explain the essential aspects of our proof and outline some of the new examples we can produce. Finally, I will mention aspects of the proof we must alter to adapt the method to higher dimensions. This work is joint with Nicos Kapouleas.

Marcelo Disconzi, Vanderbilt
 "On the Einstein equations for relativistic fluids."

Abstract: The Einstein equations have been a source of many interesting problems in Physics, Analysis and Geometry. Despite the great deal of work which has been devoted to them, with many success stories, several important questions remain open. One of the them is a satisfactory theory of isolated systems, such as stars, both from a perspective of the time development of the space-time, as well as from the point of view of the geometry induced on a space-like three surface. This talk will focus on the former situation. More specifically, we shall discuss relativistic fluids with and without viscosity, and prove a well-posedness result for the Cauchy problem. The viscous case, in particular, is of significant interest in light of recent developments in Astrophysics.

Ailana Fraser, University of British Columbia
 "Eigenvalue problems and minimal surfaces"

Abstract: Finding sharp eigenvalue bounds and characterizing the extremals is a basic problem in geometric analysis. We will describe the structure of metrics which are obtained by maximizing the first eigenvalue over all metrics on a surface (either closed or with boundary). It turns out that the extremals are related to minimal surfaces, and in some cases it is possible to use minimal surface theory to characterize the extremal metrics. This is joint work with R. Schoen.

Davi Maximo, UT Austin

"On the blow-up of four-dimensional Ricci flow singularities."

Abstract: In 2002, Feldman, Ilmanen, and Knopf constructed the first example of a non-trivial (i.e. non-constant curvature) complete non-compact shrinking soliton, and conjectured that it models a Ricci flow singularity forming on a closed four-manifold. In this talk, we confirm their conjecture and, as a consequence, show that limits of blow-ups of Ricci flow singularities on closed four-dimensional manifolds do not necessarily have non-negative Ricci curvature.

John Pardon, Stanford

"Totally disconnected groups (not) acting on three-manifolds."

Abstract: Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery--Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery--Zippin) that it suffices to rule out the case of the additive group of $p$-adic integers acting faithfully on a manifold. I will present a solution in dimension three. The proof uses tools from low-dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group.