报告1
报告题目：Fundamental gap estimate on convex domains of sphere
报告时间：2018年7月3日（星期二）09:00—09:45
报告地点：苏州大学本部精正楼211
报告人：Guofang Wei (UC Santa Barbara)
报告提要：In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space and conjectured similar results holds for spaces with constant sectional curvature. In several joint works with S. Seto, L. Wang; C. He; and X. Dai, S. Seto, we prove the conjecture for the sphere. Namely for any strictly convex domain in the unit S^n sphere, the gap is ≥ 3π^2/D^2. As in B. Andrews and J. Clutterbuck's work, the key is to prove a super log-concavity of the first eigenfunction.
报告2
报告题目：Boundary regularity for degenerate-singular Monge-Ampère equations
报告时间：2018年7月3日（星期二）10:15—11:00
报告地点：苏州大学本部精正楼211
报告人：Huaiyu Jian (Tsinghua University)
报告提要：In 1977, Cheng and Yau studied a class of Monge-Ampère Equations from affine geometry which may be singular or degenerate on the boundary. They obtained the existence, uniqueness and interior regularity for the solution. In this talk, we will discuss the boundary regularity for the solution as well as for the graph of affine hyperbolic sphere.
报告3
报告题目：On Eells-Sampson type theorems for subelliptic harmonic maps
报告时间：2018年7月3日（星期二）11:10—11:55
报告地点：苏州大学本部精正楼211
报告人：Yuxin Dong (Fudan University)
报告提要：A sub-Riemannian manifold is a manifold with a subbundle of the tangent bundle and a fiber metric on this subbundle. A Riemannian extension of a sub-Riemannian manifold is a Riemannian metric on the manifold compatible with the fiber metric on the subbundle. One may define an analog of the Dirichlet energy by replacing the L^2 norm of the derivative of a map between two manifolds with the L^2 norm of the restriction of the derivative to the subbundle when the domain is a sub-Riemannian manifold. A critical map for this energy is called a subelliptic harmonic map. In this talk, by use of a subelliptic heat flow, we establish some Eells-Sampson type existence results for subelliptic harmonic maps when the target Riemannian manifold has non-positive sectional curvature.
报告4
报告题目：On Morse index estimates for minimal surfaces
报告时间：2018年7月3日（星期二）14:00—14:45
报告地点：苏州大学本部精正楼211
报告人：Davi Maximo (University of Pennsylvania)
报告提要：In this talk we will survey some recent estimates involving the Morse index and the topology of minimal surfaces.
报告5
报告题目：A model flow for submanifolds with constant curvature
报告时间：2018年7月3日（星期二）14:55—15:40
报告地点：苏州大学本部精正楼211
报告人：Valentina Wheeler (University of Wollongong)
报告提要：One of the most basic pursuits in geometry is the understanding of shapes with least bending. In this talk, we interpret bending not as pure curvature but as a derivative of curvature (although linguistically it sounds odd, this is called the jerk), and take an energetic approach toward the analysis of shapes with least jerk. We propose a broad problem in the calculus of variations, on submanifolds with parallel mean curvature vector. As a first step, we study the problem in the geometrically mostly uninteresting case of curves in the plane. Here the gradient flow nevertheless challenges us to come up with custom-made a-priori estimates. First, we determine the set of equilibria -- circles -- through an analysis of the Euler-Lagrange equation. Then, we define a scale-invariant energy and study the flow for small enough initial energy. After some effort, we prove convergence in this energetic neighbourhood of the flow to a round circle. Apart from energy estimates, the Llojasiewicz-Simon gradient inequality makes an appearance. We quite carefully establish the gradient inequality in our setting, which although straightforward, still requires some effort. This is joint work with Ben Andrews (ANU), James McCoy (UoN) and Glen Wheeler (UOW).
报告6
报告题目：Non-concavity of the Robin ground state
报告时间：2018年7月3日（星期二）16:10—16:55
报告地点：苏州大学本部精正楼211
报告人：Daniel Hauer (The University of Sydney)
报告提要：On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this talk, I show that this is false, by analysing the perturbation problem from the Neumann case. In particular, I prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. One can conclude from this that the Robin ground state is not log-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance. This is a joint work with Ben Andrews (ANU, Canberra) and Julie Clutterbuck (Monash University).