$G(m,p,n)$ is an imprimitive complex reflection group for any positive integer $m,p,n$ with $p/mid m$. We introduce the reflection length $l_T$ and the reflection ordering $/preceq_{mp}$ on $G(m,p,n)$. We shall give some close formulae of $l_T(z)$ for any $z/in G(m,p,n)$ and some criteria for any $x,y/in G(m,p,n)$ to satisfy the relation $x/preceq_{mp}y$ when $(m,p)/in/{(1,1),(m,1),(m,m)/}$.

This study presents a new stochastic multiscale analysis approach to analyze the heat transfer performance of heterogeneous materials with random structures at different length scales. The heterogeneities of the materials are taken into account by periodic layouts of unit cells, consisting of randomly distributed inclusion dispersions and homogeneous matrix on the microscale and mesoscale. Based on the reiterated homogenization, a novel unified micro-meso-macro stochastic multiscale formulation is established and the scale gap is correlated by means of two-scale asymptotic expansions. Also, the stochastic multiscale formulae for computing the effective thermal property and temperature field are derived successively. Then, the stochastic prediction algorithm coupled with the finite element method is brought forward in details.

In this lecture, the speaker tries to present a proof of 2-dimensional Jacobian conjecture, based on several results established by the speaker mainly using the local bijectivity of Keller maps. The main contents in this lecture can be found in the paper recently posted to arxiv:1603.01867

Although there are many known K/"ahler-Einstein manifolds, there is so far no very practical method to check a given compact Fano manifold to be K/"ahler-Einstein or not.This is also true for the K/"ahler metrics with constant scalar curvatures or Calabi extremal metrics. The situation for a cohomogeneity one metrics was completely resolved.For the type III case, it was solved in my dissertation in 1992. For the remain type I and II case, the existence is equivalent to the negativity of a topological integral.The type I case was published in 2011. Therefore, the problem is reduced to check the negativity for classes of Fano manifolds. Recently, we use computer to get some insight into a few classes of type I Fano manifolds. This work is a joint work with a group of students.

The study of constant scalar curvature Kahler metrics with cone singularities is motivated from the Yau-Tian-Donaldson conjecture. We would present the recent progress in the constant scalar curvature Kahler metrics with cone singularities, including regularity, existence and uniqueness. We would also discuss the application to the constant scalar curvature Kahler metric problem.

In this lecture, the speaker tries to present a proof of 2-dimensional Jacobian conjecture, based on several results established by the speaker mainly using the local bijectivity of Keller maps. The main contents in this lecture can be found in the paper recently posted to arxiv:1603.01867

In this talk, we will present stabilized compact exponential time differencing methods (ETD) for numerical solutions of a family of gradient flow problems, which have wide applications in materials science, fluid dynamics and biological researches. These problems often form a special class of parabolic equations of different orders with high nonlinearity and stiffness, thus are often very hard to solve efficiently and robustly over large space and time scales. The proposed methods achieve efficiency, accuracy and provable energy stability under large time stepping by combining linear operator splittings, compact discretizations of spatial operators, exponential time integrators, multistep or Runge-Kutta approximations and fast Fourier transform.

In 2000, Chenciner and Montgomery proved the existence of the figure-eight solution in the planar three-body problem with equal masses by using the variational method. Since then, a number of new periodic solutions have been discovered and proven to exist. A workshop on Variational Methods in Celestial Mechanics was organized by Chenciner and Montgomery in 2003 to address the possible applications of variational method in studying the Newtonian N-body problem, while several open problems were proposed by the attending experts. The existence of the Broucke-Henon orbit is one of these open problems, which was proposed by Venturelli. Actually, he noticed that the Schubart orbit with collision is on the closure of the homology class (1, 0, 1). It is not clear if the Broucke-Henon orbit is a minimizer in the homology class (1, 0, 1).