We will talk about recent results on the proof of Cardy-Gamsa' formula on Brownian loop measure and on the chordal conformal restriction measure with random hulls. These are joint works with Yong Han and Michel Zinsmeister.

I will discuss a PDE approach to the Lp-Brunn-Minkowski inequality for p<1. The Brunn-Minkowski inequality is one of the most important inequalities in the convex geometry. After the works of Firey, Lutwak and et al., many efforts are devoted to extending the inequality to the case p<1. In particular Kolesnikov-Milman established a local Lp-Brunn-Minkowski inequality. I will discuss a proof of the global inequality using the regularity theory of Monge-Ampere equation and Leray Schauder degree theory. This is based on a joint work with Huang, Li and Liu.

In many longitudinal studies, outcomes are assessed on time scales anchored by certain clinical events. When the anchoring events are unobserved, the study timeline becomes undefined, and the traditional longitudinal analysis loses its temporal reference. We consider the analytical situations where the anchoring events are interval censored. We show that by expressing the regression parameter estimators as stochastic functionals of a plug-in estimate of the unknown anchoring event distribution, the standard longitudinal models can be modified and extended to accommodate the less well defined time scale. This extension enhances the existing tools for longitudinal data analysis. Under mild regularity conditions, we show that for a broad class of models, including the frequently used generalized mixed-effects models, the functional parameter estimates are consistent and asymptotically normally distributed with an n1/2 convergence rate.

A new multi-component diffuse interface model with the Peng-Robinson equation of state is developed. Initial values of mixtures are given through the NVT flash calculation. This model is physically consistent with constant diffusion parameters, which allows us to use fast solvers in the numerical simulation. In this paper, we employ the scalar auxiliary variable (SAV) approach to design numerical schemes. It reformulates the proposed model into a decoupled linear system with constant coefficients that can be solved fast by using fast Fourier transform. Energy stability is obtained in the sense that the modified discrete energy is non-increasing in time. The calculated interface tension agrees well with laboratory experimental data.

Over the past years, we have been working on a finite difference analog of the boundary integral equation method for elliptic and parabolic partial differential equations. We call it as the kernel-free boundary integral (KFBI) method. In this talk, I will present a proof for the validity of a simplified version of this method for the Dirichlet problem in a general domain in two or three space dimensions. Given a boundary value, the simplified method solves for a discrete version of the density of the double layer potential using a low order interface method. It produces the Shortley-Weller solution for the unknown harmonic function with second-order accuracy.

Let A = F [x, y] be the polynomial algebra on two variables x, y over an algebraically closed field F of characteristic zero. Under the Poisson bracket, A is equipped with a natural Lie algebra structure. It is proven that the maximal good subspace of A* induced from the multiplication of the associative commutative algebra A coincides with the maximal good subspace of A* induced from the Poisson bracket of the Poisson Lie algebra A. Based on this, structures of dual Lie bialgebras of the Poisson type are investigated. As by-products, five classes of new infinite-dimensional Lie algebras are obtained.

In this talk, we first review the basic results on modules, quasi modules, and $/phi$-coordinated modules for vertex algebras, and then we use examples to show the natural connections between various Lie algebras and vertex algebras.

In this talk, a semi-discrete DG method is first introduced to solve the Maxwell’s equations in three dispersive media in a uniform framework, which is described by an integral-differential equation. Accuracy of is obtained. Then it is also noted that the governing equations for three different meta-materials share a common feature. Based on this observation, a method combining the DG method in space with the CG method in time for Maxwell’s equation in meta-materials in an uniform framework is discussed. An energy identity is obtaiend and the unconditional stability is naturally reduced. Then the convergence rate of is verfied.

守恒律方程为20世纪50年代兴起的一个研究领域，此类型方程所涵盖的物理模型十分广泛，几乎所有的连续体力学的模型方程均属于这种形式，其中包含了气体、液体、弹性体、等离子体、星云等等。该领域作为数学与力学之间的一个重要枢纽十分重要，而双曲守恒律方程(组)的激波解和数值方法的研究更是科学界热门的研究课题。 本报告将回顾一下双曲守恒律数值方法的历史，并介绍与合作者一起在双曲守恒律方程（组）的数值方法的研究方面取得的一些工作，它们包括数值方法机理（局部振荡和声速点故障等）分析和高分辨自适应移动网格方法等及双曲守恒律的激波捕捉方法在计算流体力学中的应用等。

Networks come in different sizes and shapes, and it is of theoretical and practical interest to characterize aspects of networks. One important aspect is how strong the nodes in the network are associated to each other. In this paper, we propose a so-called normalized clustering coefficient (NCC) to measure the clustering strength of networks. The NCC has interesting theoretical properties and is potentially useful in many areas of network studies, e.g., network clustering, network sampling, and dynamic network analysis. Simulations and real examples will given to see how they work.