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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.

In this talk, we give the lowest order $P_1$-nonconforming triangular finite element method (FEM) for the elliptic and parabolic interface problems. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm and $L^2$ norm, respectively. Lastly, some numerical results are provided to verify the theoretical analysis.

We propose a new splitting algorithm to solve a class of semilinear parabolic PDEs with convex and quadratic growth gradients. By splitting the original equation into a linear parabolic equation and a Hamilton - Jacobi equation, we are able to solve both equations explicitly. In particular, we solve the associated Hamilton-Jacobi equation by the Hopf - Lax formula, and interpret the splitting algorithm as a Hopf-Lax splitting approximation of the semilinear parabolic PDE. We prove that the solution of the splitting scheme will converge to the viscosity solution of the equation, obtaining its convergence rate via Krylov's shaking coefficients technique and Barles-Jakobsen's optimal switching approximation. (Joint work with Shuo Huang and Thaleia Zariphopoulou)

We will talk about the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure and a continuous function there exists a convex body such that is an optimizer of the following optimization problems: .The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on , the existence of a solution is proved for a nonzero finite measure on unit sphere which is not concentrated on any hemisphere of . Another part of this paper deals with the p-capacitary Orlicz-Petty bodies.

In this talk, we will introduce the a priori method applied to the existence theory of elliptic PDEs in R^n and S^n. There are some open problems in this subject, especially for the degenerate elliptic PDEs. At last, we prove the existence theorem of the non-symmetric solutions of the homogeneous Lp dual Minkowski problem, which is a joint work with Yong Huang and Lu Xu

We study testing high-dimensional covariance matrices under a generalized elliptical model. The model accommodates several stylized facts of real data, including heteroskedasticity, heavy-tailedness, asymmetry, etc. We consider the high-dimensional setting where the dimension p and the sample size n grow to infinity proportionally, and establish a central limit theorem for the linear spectral statistic of the sample covariance matrix based on self-normalized observations. The central limit theorem is different from the existing ones for the linear spectral statistic of the usual sample covariance matrix.