Hemivariational inequalities are nonsmooth and nonconvex problems. They arise in a variety of applications in sciences and engineering. For applications in mechanics, through the formulation of hemivariational inequalities, problems involving nonmonotone, nonsmooth and multivalued constitutive laws, forces, and boundary conditions can be treated successfully. In the recent years, substantial progress has been made on numerical analysis of hemivariational inequalities. In this talk, a summarizing account will be given on recent and new results on the numerical solution of hemivariational inequalities with applications in contact mechanics.

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.