The Southeast Geometry Seminar (SGS) is a semiannual series of one day event sponsored by:
The organizers are: Vladimir Oliker (Emory) and Gideon Maschler (Emory), Mohammad Ghomi and John McCuan (Georgia Tech), and Gilbert Weinstein (UAB).
I will discuss two kinds of symmetry for surfaces which both generalize "being invariant under isometries fixing a geodesic." The primary focus of the talk will be the reduction of the most general of these symmetries to the less general ones when the surface is compact. This reduction allows us to apply a recent classification result for these surfaces, which we will also describe briefly.
We will present a recent result of microscopic convexity principle for geometric fully nonlinear eqautions. The work is an accumultion of joint works with B. Bian, L. Caffarelli and Ma in recent years. We will also discuss some applications. We prove a sphere theorem for a class of Weingarten hypersurfaces extending some classical results, a uniqueness theorem for Kahler metrics on compact manifolds with nonnegative orthogonal bisectional curvature.
Nonlinear equations appear naturally in differential geometry, for example when studying the cutrvature of hypersurfaces in Euclidean space. Indeed, many of the fundamental advances in the theory of partial differential equations in the last century were inspired by considering very concrete problems in geometry. Over the last five or ten years there has been considerable interest in certain fully nonlinear equations in conformal geometry. In this talk I will discuss some geometric applications of these equations, and some open questions about existence and regularity.
For smooth convex bodies in ℝn, there are two classical SL(n) invariant notions of surface area: affine surface area and centro-affine surface area. Recently, these definitions have been extended to general convex bodies. We describe applications of these affine surface areas to problems of polytopal approximation and give further geometric interpretations. We also present an SL(n) invariant analogue of Hadwiger's classical characterization theorem
Joint work with Matthias Reitzner, Technische Universität Wien.
I will discuss the Calabi-Yau equation on a compact symplectic manifold with a compatible almost complex structure. This is an analogue of the equation on Kahler manifolds solved by S. T. Yau thirty years ago. I discuss some progress on a conjecture of Donaldson about this equation and how it relates to questions in symplectic geometry. Part of this is joint work with V. Tosatti and S. T. Yau.
Let n1 and n2 be the indexes of refraction of two homogeneous and isotropic media I and II, respectively. Suppose that from a point O inside medium I light emanates with intensity f(x) for x ∈ Ω. We seek a refracting surface ℜ parameterized by ℜ = { f(x) : x ∈ Ω }, separating media I and II, and such that all rays refracted by ℜ into medium II have directions in Ω* and the prescribed illumination intensity received in the direction m ∈ Ω* is f*(x). We prove that the surface ℜ exists and is unique up to dilations.
This is joint work with Qingbo Huang.