Mathematical Sciences Research Institute

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Geometric Evolution Equations and Related Topics August 14, 2006 to May 25, 2007
Organizers Bennett Chow, Panagiota Daskalopoulos, Gerhardt Huisken, Peter Li, Lei Ni, Gang Tian
Description The focus of the program is on geometric evolution equations, function theory, and related elliptic and parabolic equations. Geometric flows have been applied to a variety of geometric, topological, analytical and physical problems. Linear and nonlinear elliptic and parabolic partial differential equations have been studied by continuous, discrete and computational methods. Geometric flows evolve geometric structures by diffusive (heat type) equations in geometric form. Many such flows are driven by some form of curvature such as the mean, inverse mean, Gauss, and Willmore flows of submanifolds, the Ricci, Kähler-Ricci, and Calabi flows of manifolds, the Yang-Mills and Hermitian-Einstein flows of connections and metrics on vector bundles, and the Yamabe and other conformal flows of metrics. Other flows include the harmonic map heat flow and the porous medium flow. Each of these flows seeks to evolve the corresponding geometric structures to canonical geometric structures. The analysis of geometric flows is related to the study of the heat equation and function theory, nonlinear parabolic equations, computational methods, and the discrete heat equation on graphs. The analysis of singularity formation is one of the most fundamental topics in geometric flows. The study of singularities is part of a broader program to understand global existence and convergence. Global solutions are sought either under geometric conditions on the initial data, after surgery (such as in the Ricci flow and mean curvature flow), or for weak solutions. There are numerous relations between geometric flows and other areas of mathematics and science. Mean curvature flow, which is the gradient flow for the area functional, and its variants are related to material science. Gauss curvature flow has been used to model the wearing of stones and other objects. Inverse mean curvature has been applied to solve fundamental problems in general relativity, such as the Penrose inequality. Ricci flow has been studied as a means to solve the Poincare and Geometrization Conjectures.The Kähler-Ricci and Hermitian-Einstein flows have deep applications to algebraic geometry. Recent developments in all of these areas make the field of geometric flows ripe for further striking developments. The Fall program will include the following topics: Function theory and the analysis of the Laplace and heat equations on manifolds: harmonic function theory on complete Riemannian manifolds, spectrum of the Laplacian operator, spaces of holomorphic functions on complete Kähler manifolds, partial differential equations on complete manifolds and their relation with the geometry of complete Riemannian and Kähler manifolds. Nonlinear elliptic and parabolic methods in geometry and analysis including Kähler-Einstein metrics, Yamabe problem, and the porous medium equation. The Spring program will include geometric flows, such as the mean, inverse mean, Gauss curvature and Willmore flows of submanifolds, Ricci/Kähler-Ricci and Calabi flows of manifolds, Yang-Mills and Hermitian-Einstein flows of connections and metrics on vector bundles, and Yamabe and other conformal flows of metrics. Applications include the Geometrization Conjecture (in the process of verification), Penrose inequality, uniformization theorems in Kähler geometry, existence of canonical metrics, connections and maps such as Einstein, Hermitian-Einstein, constant scalar curvature, and extremal metrics, Yang-Mills connections, and harmonic maps. Techniques include the geometric theory of a priori estimates, singularity analysis, surgery techniques, weak solutions, global existence and convergence. The field of geometric flows has had a rapid recent development and connections between previously less related areas and flows are being developed. The program will bring together researchers working on different geometric flows and allow for cross-fertilization of ideas and techniques from various flows and will enable participants to pursue such connections between various flows and areas of geometric analysis.

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Programmatic Workshops
September 08, 2006 - September 09, 2006 Connections for Women: Geometric Analysis and Nonlinear Partial Differential Equations
September 11, 2006 - September 15, 2006 Introductory Workshop on Geometric Flows and Function Theory in Real and Complex Geometry
October 23, 2006 - October 27, 2006 Analytic and Computational Aspects of Elliptic and Parabolic Equations
March 12, 2007 - March 16, 2007 Geometric Evolution Equations
March 16, 2007 - March 17, 2007 Recent Developments in Numerical Methods and Algorithms for Geometric Evolution Equations