We discuss an arbitrary-order discontinuous Galerkin method for second order elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error estimates for the energy norm and for the L2 norm. The accuracy and the efficiency of the method up to order six on several polygonal meshes are illustrated by a set of benchmark problems. The application of this method to plate bending problem will also be addressed. This is a joint work with Ruo Li, Ziyuan Sun and Zhijian Yang.

In this talk, we discuss an optimal investment and consumption problem with exponential utility function in a financial market where the asset prices follows a cointegrated model. After applying the dynamic programming method, we derive a Hamilton-Jacobi-Bellman (HJB) equation, then we obtain an optimal investment and consumption strategies and the corresponding value function in a closed form. A verification theorem is proved to demonstrate that under certain growth conditions the solution of the HJB equation is indeed the one of our original problem.

The talk will deal with the following question. Given a homeomorphism f of a surface S and a set X (of S) of fixed points of f, can one find a continuous path f_t (t/in [0,1])of homeomorphisms of S joining the identity to f, so that f_t fixes X pointwise for every t? If such a path (f_t) exist and X is maximal, then a theorem of Le Calvez allows to describe the dynamics of f. This is a joint work with Sylvain Crovisier (Université Paris-Sud) and Frédéric Le Roux (Université Pierre et Marie Curie - Paris)

The coefficients of asymptotic expansion of Bergman kernel on Kahler manifolds give important geometric information. We show that they could be expressed in a compact form as a summation over strongly connected graphs. The relationship to deformation quantization and heat kernel will be discussed.

题目： Free Fields and Affine Lie Superalgebras of Type A报告人：郜云 教授地点：致远楼108室时间：2018年4月19日 16:00-17:00报告人简介：郜云， 加拿大 York 大学教授，上海大学理学院核心数学研究所所长、博士生导师；德国洪堡学者，国家海外杰出青年基金获得者欢迎广大师生参

In this talk, we will discuss the behavior of the Chern-Ricci flow (CRF) on Hermitian manifolds. The Chern-Ricci flow is an evolution equation for Hermitian metrics on complex manifolds. In particular, we investigate the Chern-Ricci flow on Inoue surfaces which are non-K/"ahler compact complex surfaces of type Class VII. We show that, after an initial conformal change, the flow always collapses the Inoue surface to a circle at infinite time, in the sense of Gromov-Hausdorff. Similar convergence result also holds on the Oeljeklaus-Toma(OT) manifolds, an analog of Inoue surface.

Medical imaging is the technique and process of visualizing the anatomy of a body for clinical analysis and medical intervention, as well as the function of some organs and tissues. However, the reconstructed image in general suffers from the severe artifacts due to the ill posed nature of underlying inverse problem. In this talk, I will briefly introduce some related topics in the inverse problem in medical imaging based on my works. One is the mathematical analysis of the inverse problem in quantitative susceptibility mapping to present the existence and uniqueness, and to characterize the streaking artifacts due to the ill posed nature of the inverse problem. The other is the edge driven wavelet frame based image restoration model which is designed to restore/enhance the key features in a given image.

Motivated by applications in image processing, we study asymptotic behavior for the level set equation of power mean curvature flow as the exponent tends to 0 or to infinity. When the exponent is vanishing, we formally obtain a fully nonlinear singular equation that describes the motion of a surface by the sign of its mean curvature. We justify the convergence by providing a definition of viscosity solutions to the limit equation and establishing a comparison principle. In the large exponent case, the limit equation can be characterized as a stationary obstacle problem involving 1-Laplacian when the initial value is assumed to be convex.