High order nonlinear weighted essentially non-oscillatory finite difference (WENO) schemes, which have the capability of capturing shocks essentially non-oscillatory while re-solving small scale structures efficiently, are popular for solving hyperbolic conservation laws. However, the WENO scheme is fairly complex to implement, computationally expen-sive and too dissipative for certain classes of problems. A natural way to alleviate some of these difficulties is to construct a hybrid scheme conjugating a nonlinear WENO scheme in the non-smooth stencils with a linear method in the smooth stencils. In this talk, I will briefly introduce the difficulties and recent development in the hybrid schemes.

In this talk, we first give a brief introduction to the optimal transport problem, and then its extension to nonlinear case with applications in geometric optics. Last, we introduce some recent results on the optimal partial transport problem, which is based on joint work with Shibing Chen (USTC) and Xu-Jia Wang (ANU).

We will discuss a class of Monge-Ampere type equations defined on the unit hypersphere, which are related to the Orlicz-Brunn-Minkowski theory in modern convex geometry. These equations are fully nonlinear partial differential equations, and could be degenerate or singular in different cases. We will talk about some recent results about the existence and non-uniqueness of solutions to these equations.

报告人将介绍在高维唯一性问题领域的最新工作，包括三个方面：一是对于一般的可能线性退化的亚纯映射，在一个新类型的条件下得到的唯一性结果；二是分担 2n+2 个超平面的唯一性结果；三是分担较少超曲面的退化性和唯一性结果。

A robust numerical method via the shape derivatives and low-rank approximation is developed for computations of three-dimensional Maxwell's equations with random interfac-es. Based on a shape calculus, we estimate the statistical moments of the stochastic Maxwell equations in terms of perturbation magnitude. In order to capture the oscillations with high resolution near the interface, we adopt the adaptive edge element with third order polynomi-als to solve the deterministic equations approximating the expectation. For the second mo-ment, an efficient low-rank approximation based on pivoted Cholesky decomposition is pro-posed to compute the two-point correlation function to approximate the variance of stochastic Maxwell's equations. Numerical experiments are presented to illustrate our theoretical results.

We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure (M; g). This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex structure J has small energy(depending on the norm |/nabla J|), then the flow exists for all time and converges to a Kaehler structure. We also prove that there is a finite time singularity if the initial energy is sufficiently small but there is no Kaehler structure in the homotopy class. A main technical tool is a version of monotonicity formula, similar as in the theory of the harmonic map heat flow. We also construct an almost complex structure on a flat four tori with small energy such that the harmonic heat flow blows up at finite time with such an initial data.

We shall first give a brief introduction for Alexandrov geometry, which is a class of singular metric spaces with curvature bounded from below by triangle comparisons, and then we introduce a result about regularity of harmonic maps from an Alexandrov space to a compact Riemannian manifold, which is based on the joint works with Prof. Huabin Ge and Prof. Wenshuai Jiang.

We solve the Gursky-Streets equations with uniform C^1,1 estimates for . An important new ingredient is to show the concavity of the operator which holds for all . Our proof of the concavity heavily relies on Garding's theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform C^1,1 a priori estimates for all the cases . Moreover, we establish the uniqueness of the solution to the degenerate equations for the first time.