Workshop

Geometric flows have been applied to a variety of geometric, topological, analytical and physical problems. Such flows include 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. SCHEDULE: Monday March 12 9:30 am -- 10:30 am Burkhard Wilking Manifolds with positive curvature operators are space forms 11:00 am -- 12:00 pm Natasa Sesum Title TBA 2:00 pm -- 3:00 pm Felix Schulze Nonlinear evolution by mean curvature and isoperimetric inequalities 4:10 pm -- 5:00 pm MSRI/Evans Talk 60 Evans Hall; University of California, Berkeley Robert Bryant Finsler surfaces of constant curvature Tuesday March 13 9:30 am -- 10:30 am Klaus Ecker Title TBA 11:00 am -- 12:00 pm Rafe Mazzeo CMC foliations and area-minimizing submanifolds in asymptotically hyperbolic spaces 2:00 pm -- 3:00 pm Weiyong He Title TBA 3:30 pm -- 4:30 pm Simon Brendle Title TBA Wednesday March 14 9:30 am -- 10:30 am John Lott On type-III Ricci flow solutions 11:00 am -- 12:00 pm Yuanlong Xin Convexity, Rigidity and Mean Curvature Flow 2:00 pm -- 3:00 pm Dan Knopf Title TBA 3:30 pm -- 4:30 pm Qi Zhang A Li-Yau type inequality for the conjugate heat equation under Ricci flow Thursday March 15 9:30 am -- 10:30 am Todd Oliynyk Title TBA 11:00 am -- 12:00 pm Jiayu Li Title TBA 2:00 pm -- 3:00 pm Xiuxiong Chen Title TBA 3:30 pm -- 4:30 pm Reto Mueller Differential Harnack Inequalities and Monotone Quantities Friday March 16 9:30 am -- 10:30 am Jan Metzger Title TBA 11:00 am -- 12:00 pm Andre Neves Convergence of inverse mean curvature flow in asymptotically hyperbolic manifolds Talks from Recent Developments in Numerical Methods and Algorithms for Geometric Evolution Equations will be scheduled in the afternoon and the next day (Saturday). _______________________________________________________________ Abstracts: Felix Schulze Abstract: Evolving smooth, compact hypersurfaces in R^{n+1} with normal speed equal to a positive power k of the mean curvature improves a certain 'isoperimetric difference' for k >= n-1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n <= 7. Extending this to complete, simply connected 3-dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds. Andre Neves Abstract: I will show that inverse mean curvature flow does not have the asymptotic convergence needed to prove a Penrose-type inequality for asymptotically hyperbolic 3-manifolds. A block of rooms has been reserved at the Hotel Durant. Please mention the workshop name and reference the following code when making reservations via phone, fax or e-mail: K70000. The cut-off date for reservations is February 9, 2007. A block of rooms has been reserved at the Rose Garden Inn. Reservations may be made by calling 1-800-992-9005 OR directly on their website. Click on Corporate at the bottom of the screen and when prompted enter code MATH (this code is not case sensitive). By using this code a new calendar will appear and will show MSRI rate on all room types available. The following Group Code should be entered in the "Comment" field when booking through our Website. Group Code = CGMS38. Please note: The Original Queen room type does not have an in-room T1 connection, but we do have WIFI available in our main lobby area. The cut-off date for reservations is February 9, 2007. The funding deadline has passed. Late funding requests will be considered only if additional funds become available.