We consider long time behavior of a given smooth convex embedded closed curve evolving according to a non-local curvature flow, which arises in a Hele-Shaw problem and has a prescribed rate of change in its enclosed area A (t), i.e. , where is given. Specifically, when the enclosed area expands at any fixed rate, i.e. or decreases at a fixed rate one has the round circle as the unique asymptotic shape of the evolving curves; while for a sufficiently large rate of area decrease, one can have n-fold symmetric curves (which look like regular polygons with smooth corners) as extinction shapes (self-similar solutions).

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.