The study of Dynamical Systems is mainly concerned with orbit structure, specifically long term or asymptotic behavior, for maps or flows. Uniformly Hyperbolic systems are standard examples of complex or chaotic systems. However, uniformly hyperbolic systems are not dense in the space of all dynamical systems. After that people tried to know the world beyond uniform hyperbolicity for which there are many open questions proposed by Bowen, Palis etc. In this talk we will introduce some progress on Bowen's one question to search specification-like properties and statistical properties and Palis SRB conjecture to search the existence of SRB measures.

The Algebraic Birkhoff Factorization (ABF) of Connes and Kreimer gives an algebraic formulation of the renormalization process in quantum field theory. Their ABF is an factorization of an algebra homomorphism from a Hopf algebra to a Rota-Baxter algebra. This algebraic formulation facilitates the mathematical study in renormalization and allows the renormalization method to be applied to problems in mathematics.

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.