# Program

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All lectures to be held in Center 105. Click here for a pdf map.

 Saturday morning, 2/11/17 9:00am-9:45am Meet at Lobby of the Hotel (Del Mar Inn, otherwise you need to let us know ahead of time so that proper arrangement can be made. Many Hotels also provide shuttle service to campus. Ask about it the night before) for possible ride to campus. Walking is also possible, but takes longer. 10:00am-10:50am Registration. Coffee and snacks served 10:50am-11:00am Welcome and a brief introduction 11:00am-11:50am Williams Minicozzi : Level set method for motion by mean curvature Saturday afternoon, 2/11/17 1:30pm-2:20pm Fang-Hua Lin: Revisit Optimal Partition of Dirichlet Eigenvalues 2:20pm-2:50pm Break, refreshments served 2:50pm-3:40pm Lu Wang : Asymptotic structure of self-shrinkers 3:40pm-4:10pm Break, refreshments served 4:10pm-5:00pm Michael Wolf: Sheared Pleated surfaces and Limiting Configurations for Hitchin's equations 5:30pm Meet for ride to restaurant at APM building 1st floor 6:15pm Conference dinner at (cost $40per person)Jasmine Sea Food Sunday morning, 2/12/17 8:15am-8:30am Meet at lobby of hotels for ride to campus. Walking is also possible, but takes longer. 8:40am-9:10am Coffee and snacks 9:10am-10:00am André Neves : Weyl Law for volume spectrum 10:00am-10:25am Break, refreshments served 10:25am-11:15am Joel Spruck : Convexity of complete translating solitons to the mean curvature flow in$R^3$with nonnegative mean curvature 11:15am-11:40am Break, refreshments served 11:40am-12:30pm Li-Sheng Tseng : Odd sphere bundles and symplectic manifolds ## Abstracts: Fang-Hua Lin (NYU). Revisit Optimal Partition of Dirichlet Eigenvalues Abstract: Let$\Omega$be a bounded domain in$R^n$, and$m$a positive interger. We are interested in the following problem: Find a partition of$\Omega$into$m$mutually disjoint subsets$\Omega_j$,$j= 1, 2,..., m$, such that the sum of the Dirichlet first eigenvalues of$\Omega_j$'s is minimized among all possible partitions of$\Omega$. In this talk I shall review some earlier results and recent progress on this problem. Williams Minicozzi (MIT). Level set method for motion by mean curvature Abstract:Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to a degenerate elliptic nonlinear pde. A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. This is joint work with Toby Colding. André Neves (Chicago). Weyl Law for volume spectrum Abstract:Gromov in the 80’s introduced the notion volume spectrum and conjectured that it obeys a Weyl asymptotic Law. I will talk about its recent proof, jointly with Liokumovich and Marques, and how the volume spectrum related with some other well known open problems in Geometry. Joel Spruck (Johns Hopkins). Convexity of complete translating solitons to the mean curvature flow in$R^3$with nonnegative mean curvature. Abstract: We prove that any complete immersed two sided mean convex translating soliton$\Sigma \subset R^3$for the mean curvature flow is convex. As a corollary it follows that any entire mean convex graphical translating soliton in$R^3$is the axisymmetric “bowl soliton''. We also show that if the mean curvature of$\Sigma$tends to zero at infinity, then$\Sigma$can be represented as an entire graph and so is the bowl soliton . Locally strictly convex translating solitons defined over strips (the only other nontrivial solutions) are both interesting and complicated. They can only exist on strips of width$2R>\pi$. For$2R=\pi$, the standard grim cylinder is the unique solution while for$2R<\pi\$ there is no solution. This is joint work with Ling Xiao.

Li-Sheng Tseng (UC, Irive). Odd sphere bundles and symplectic manifolds

Abstract: I will motivate the consideration of a special class of odd dimensional sphere bundles over symplectic manifolds. These bundles give a novel topological perspective for symplectic geometry. In particular, the symplectic A-infinity algebra recently found by Tsai-Tseng-Yau turns out to be equivalent to the standard de Rham differential graded algebra of forms on the sphere bundles. The bundle picture also points to an intersection theory of coisotropic/isotropic chains on symplectic manifolds. This talk is based on joint work with Hiro Tanaka.

Lu Wang (Wisconsin). Asymptotic structure of self-shrinkers

Abstract: Self-shrinkers are singularity models for mean curvature flow. In this talk, I will show that each end of a noncompact self-shrinker in the Euclidean three-space of finite topology is smoothly asymptotic to a regular cone or a round cylinder.

Michael Wolf (Rice). Sheared Pleated surfaces and Limiting Configurations for Hitchin's equations

Abstract: A recent work by Mazzeo-Swoboda-Weiss-Witt describes a stratum of the frontier of the space of SL(2,C) surface group representations in terms of 'limiting configurations' which solve a degenerated version of Hitchin's equations on a Riemann surface. We interpret these objects in terms of the hyperbolic geometric objects of shearings of pleated surfaces and accompanying decorated real trees. (Joint with Andreas Ott, Jan Swoboda, and Richard Wentworth).