All lectures to be held in Center 105. Click here for a pdf map.
| Saturday morning, 2/23/13 | |
| 9:00am-9:45am | Meet at Lobby of the Hotel for possible ride to campus. Walking is also possible, but takes longer. |
| 10:00am-11:00am | Registration. Coffee and snacks served |
| 11:00am-11:50am | Alice Chang: Boundary regularity of Bach flat metrics |
| Saturday afternoon, 2/23/13 | |
| 1:30pm-2:20pm | Pengfei Guan: Geometric Inequalities and Curvature Flows |
| 2:20pm-2:50pm | Break, refreshments served |
| 2:50pm-3:40pm | Aaron Naber: Classification of Tangent Cones and Lower Ricci Curvature |
| 3:40pm-4:10pm | Break, refreshments served |
| 4:10pm-5:00pm | Steven Zelditch: Grauert tubes and complex geometry of eigenfunctions |
| 5:30pm | Meet for ride to restaurant at APM building 1st floor |
| 6:15pm | Conference dinner at Pearl |
| Sunday morning, 2/24/13 | |
| 8:15am-8:30am | Meet at lobby of hotels for ride to campus. Walking is also possible, but takes longer. |
| 8:40am-9:10am | Light breakfast and snacks |
| 9:10am-10:00am | Hubert Bray: On Dark Matter, Galaxies, and the Large Scale Geometry of the Universe |
| 10:00am-10:25am | Break, refreshments served |
| 10:25am-11:15am | Gerard Besson: Differentiable rigidity, Ricci and scalar curvature |
| 11:15am-11:40am | Break, refreshments served |
| 11:40am-12:30pm | Gang Tian: Stability and Einstein metrics. |
S.Y. Alice Chang (Princeton). Boundary regularity of Bach flat metrics
Abstract: In this talk, I will report some on going joint work with Yuxin Ge, Paul Yang and myself on a regularity problem of Bach flat metrics on 4-manifolds with boundary. I will discuss the set up of a matching boundary condition; an $\epsilon$ regularity result, and as an application some compactness result for metrics of conformal compact Einstein 4-manifolds.
Gerard Besson (Grenoble). Differentiable rigidity, Ricci and scalar curvature
Abstract:
Hubert Bray (Duke). On Dark Matter, Galaxies, and the Large Scale Geometry of the Universe
Abstract: More than 95% of the present day curvature of the universe is not a result of regular baryonic matter represented by the periodic table of elements. About 73% is well described by a geometrically natural cosmological constant, also referred to as dark energy, which results in a very small amount of curvature uniformly spread throughout the universe. We will explore the possibility that the remaining 23%, commonly referred to as dark mater, could also be explained very naturally from a geometric point of view. We propose geometric axioms which result in the Einstein-Klein-Gordon equations and model dark matter with the resulting scalar field. We will present simulations and models of the resulting ``wave dark matter'' theory (aka scalar field dark matter and boson stars) in dark matter dominated systems including spiral galaxies, elliptical galaxies, and dwarf spheroidal galaxies. We will compare these predictions to the observed data and photos of actual galaxies.
Pengfei Guan (McGill). Geometric Inequalities and Curvature Flows
Abstract: We discuss relationship of curvature flows of hypersurfaces and isoperimetric type inequalities. The basic idea is to design certain flow so that the isoperimetric ratio of the corresponding geometric quantities is monotone. We explore two types of such flows for starshaped domains in space forms: mean curvature type and inverse mean curvature type. The longtime existence and convergence will be used to prove isoperimetric type inequalities.
Aaron Naber (MIT). Classification of Tangent Cones and Lower Ricci Curvature
Abstract: We consider limit spaces (M_i,g_i,p_i)->(X,d,p), where the spaces M_i are noncollapsed and have Ricci curvature uniformly bounded from below. In this case we study the set TC(p) of metric spaces which consists of the possible tangent cones at p, and give a classification result which says exactly which subsets of all metric spaces can arise as TC(p) for some such limit. We use this to build new examples of limit spaces with particularly degenerate behaviors. In particular we show limit spaces cannot be stratified based on their tangent cones, and that there exists a limit space for which there are even nonhomeomorphic tangent cones at a point. This is joint work with Toby Colding.
Gang Tian (Princeton/Beijing). Stability and Einstein metrics
Abstract:
Steve Zelditch (Northwestern). Grauert tubes and complex geometry of eigenfunctions
Abstract: My talk is about eigenfunctions of the Laplace operator on real analytic Riemannian manifolds (M, g). Such M can be complexified and the metric g induces a Kahler metric and potential rho on the complexification M_C. The eigenfunctions analytically continue to a tube \rho < c, which is a strictly pseudoconvex domain in M_C. The Grauert tube may be identified with `phase space' T^* M. It turns out that many difficult conjectures about nodal sets, L^p norms and quantum ergodic problems in the real domain become simpler in the complex domain, i.e. in phase space. The results have applications back in the real domain. We survey some of the results.