题目：Approximating Signals from Random Sampling in a Reproducing Kernel Subspace of Homogeneous Type报告人：冼军 教授 （中山大学）地点：宁静楼108室时间：2019年06月21日16:00-17:0

For the numerical simulation of Lagrangian multi-material radiation hydrodynamic problems, a challenging problem is to construct discrete schemes to solve energy diffusion equations with discontinuous coefficients on Lagrangian distorted meshes. This talk will describe briefly the physical background and some key numerical issues related to cell-centered finite volume schemes. Then some finite volume schemes satisfying maximum principle for diffusion equations on general meshes are constructed with a new way. Moreover, some numerical results will be presented to show the performance of our schemes.

This talk focuses on recent progress regarding the collapsing theory of Einstein spaces. Specifically, the talk consists of the following primary parts: (1) We will introduce some regularity and structure theorems for higher dimensional collapsing manifolds with Ricci curvature bounds which works in very general settings. (2) This talk is also concerned with the geometry of collapsed Ricci-flat Kähler manifolds. Besides exhibiting new collapsing Einstein families, we will establish the correspondence between complex structures (Type II) degeneration and the metric collapsing.

In 1950’s, Nash-Kuiper built up the C^1 isometric embedding for any surface into $/mathbb{R}^3$, this can be viewed as analysis side of isometric embedding. On the other hand, there is obstruction for the existence of $C^2$ isometric embedding of surface into $/mathbb{R}^3$ known since Hilbert, which reflects the geometry flavor of isometric embedding. What’s happening from $C^1$ to $C^2$ (from analysis to geometry)? We will present our partial progress along this direction. The talk will be accessible to audience with basic knowledge of analysis and differential geometry.

The subspace of symmetric tensors and the subspace of anti-symmetric tensors are two natural reducing subspaces of tensor product A⊗I+I⊗A and A⊗A for any bounded linear operator A on a complex separable Hilbert space H. We show the set of operators A such that these two subspaces are the only (nontrivial) reducing subspaces of A⊗I+I⊗A is a dense G_δ set in B(H). This generalizes Halmos’s theorem that the set of irreducible operators is a dense G_δ set in B(H). The same question for A⊗A is still open.

In this presentation, we will give some recent progress on the geometry of compact complex manifolds with RC-positive tangent bundles including confirming a long-standing open problem of Yau on rational connectedness and various rigidity theorems of harmonic maps and holomorphic maps.

For a complex Hilbert space H, the d-copy tensor product of H is denoted by H^⊗d. For a class of tensor products of operators on H^⊗d which are invariant under a subgroup of the permutation group of d element, we identify their reducing subspaces. These reducing subspaces are formally (or implicitly) known through Schur-Weyl duality in the group representation theory where finite dimensional vectors spaces and the invertible similarity are general used. We state these results in the operator theoretic framework which deals with infinite dimensional complex Hilbert spaces and uses the unitary similarity. We explicitly display some of these reducing subspaces. Most importantly we initiate the investigation of the question for which operator these reducing subspaces are minimal.

We study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any integer k>0, as the genus g goes to infinity, the smallest k-th eigenvalue of a Riemann surface in any thick part of moduli space of Riemann surfaces of genus g is uniformly comparable to 1/g^2 in g. This is a joint work with Yuhao Xue.