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In this talk we provide a new method to study global dynamics of planar quasi homogeneous differential systems. We first prove that all planar quasi-homogeneous polynomial differential systems can be translated into homogeneous differential systems and show that all quintic quasi-homogeneous but non-homogeneous systems can be reduced to four homogeneous ones. Then we present some properties of homogeneous systems, which can be used to discuss the dynamics of quasi-homogeneous systems.

This talk concerns with a reaction-diffusion system modeling the population dynamics of two predators and one prey with nonlinear prey-taxis. We first investigate the global existence and boundedness of solution for the general model. Then we study the global stabilities of nonnegative spatially homogeneous equilibria for an explicit system with type I functional responses and density-dependent death rates for the predators and logistic growth for the prey. Moreover, the convergence rates are established.

In this talk, we will talk about the rotational symmetry of gradient steady ricci solitons with linear dacay. Perelmann conjectured that any 3-d nonflat noncollapsed steady ricci solitons must be rotationally symmetric. This conjecture has been solved by Brendle aroung 2012. Recently, we generalize this result to noncollapsed steady ricci solitons with nonnegative curvature operator in high dimensions in addition that the scalar curvature has linear decay. This is a joint work with Prof. Xiaohua Zhu.

The 1977’ Acta paper by Lawson-Osserman studied the Dirichlet problem for minimal surfaces of high codimensions. Several astonishing results essentially distinct from the case of codimension 1 were obtained there. In particular, they found Lipschitz but non-C1 solutions to the problems associated to Hopf maps between unit spheres. Recently we made systematic developments and discovered certain interesting new phenomena on the existence, non-uniqueness, non-minimizing and minimizing properties of solutions to related Dirichlet problems. This talk is based on joint works with Xiaowei XU and Ling YANG.

In this talk i will present a general theory of mathematical modeling for irreversible processes. By mathematical modeling, I mean to establish some relations among the apparently unrelated time-space functions characterizing a given non-equilibrium thermodynamic system. This theory is initiated with an observation that many classical mathematical models (hyperbolic PDEs) share certain common properties. Because of this, it is natural to require the constructed models to possess the properties when modeling an irreversible phenomenon with PDEs. Models constructed with this theory are hyperbolic balance laws and fulfill some fundamental requirements.

Recently there is some progress on the rationality of compact Kahler manifolds with positive holomorphic sectional curvature, for example the work of Heier-Wong and Xiaokui Yang. In this talk, we present a differential-geometric result on such manifolds, and exhibit new examples on Kahler and Hermitian manifolds.

Minkowski problem is one of a central problem in convex geometry. In this talk, we will present our recent work on Lp Minkowski problem for capacity.

Consider weak solutions of the instationary Navier-Stokes system in a three-dimensional bounded smooth domain $/Omega$. It is well known that any solenoidal initial value $u_0$ in $L^2(/Omega)$ with a vanishing normal component on the boundary admits a global in time weak solution. Moreover, if $u_0 /in H^1$ or even only $u_0 /in /mathcal{D}(A^{1/4})/subset L^3$, where $A =-P/Delta$ denotes the Stokes operator, then $u_0$ admits a unique local in time regular (strong) solution in Serrin’s class $L^s(0,T;L^q (/Omega))$ where $2/s + 3/q = 1$ for some $T = T(u_0)/leq/infty$.