Using quantum differential operators, we construct a super representation for the quantum linear supergroup on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a submodule isomorphic to the regular representation of the supergroup. In this way, we obtain a new presentation of the supergroup by a basis together with explicit multiplication formulas of the basis elements by generators.

The Willmore conjecture states that the Clifford torus minimizes uniquely the Willmore energy /int (H^2+1) dM among all tori in S^3, which is solved recently by Marques and Neves in 2012. For higher genus surfaces, it was conjectured by Kusner that the Lawson minimal surface, /xi_{m,1}: M-->S^3, minimizes uniquely among all genus m surfaces in S^n. The conjecture reduces to the Willmore conjecture for tori if m=1, since /xi_{1,1} is the Clifford torus. In this talk, we will prove this conjecture under the assumption that the (conformal) surfaces in S^n have the same conformal structure as /xi_{m,1}.

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.