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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).

We discuss unique solvability of the equality-constraint quadratic programming problem, establish a class of preconditioned alternating variable minimization with multiplier (PAVMM) methods for iteratively computing its solution, and demonstrate asymptotic convergence property of these PAVMM methods. We also discuss an algebraic derivation of the PAVMM method by making use of matrix splitting. Numerical results about a few image recovering problems show the effectiveness of this class of methods.

A health emergency is a situation that poses an immediate risk to health and life and requires urgent intervention to prevent it from getting worse. An efficient clinic workflow and adequate provisioning of various types of resources in an emergency department (ED) help improve the timeliness of emergency care services. Such resources include doctors, physician assistants, registered nurses, registration clerks, examination beds, CT machines, etc. This presentation proposes a simulation framework that supports the analysis of optimal provisioning of resources that ensures timely care service and meanwhile incurs no excessive cost from unnecessary resources acquisition. A stochastic timed Petri net based healthcare workflow and resource modeling technique is presented.

It is known that SYZ fibration of Calabi-Yau plays a key role in studying mirror symmetry。A natural question is to consider fibration structure in other manifolds with special holonomy。In this talk，we will report some related work for G2-manifolds。

Complete embedded constant mean curvature (CMC) surfaces of fixed, finite topology comprise a finite-dimensional moduli space. In case of coplanar CMC surfaces of genus 0 with k ends, this moduli space can be identified with the space of holomorphic immersions from the plane to the 2-sphere whose Schwarzian derivative is a polynomial with degree depending on k. Can you guess which families of coplanar CMC surfaces correspond to the polynomials 0, or 1, or z?

In this talk, we consider the dual Minkowski problem, proposed by Huang-Lutwak-Yang-Zhang (Acta 2016), for the planar case and the index $q>0$, without any symmetry assumptions. If the prescribed measure has a density which is bounded between two positive constants, we show the existence of solutions to the problem. If the density is smooth, we show the smoothness of the solutions. Some other applications of our method will also be discussed. This is a joint work with Dr. Shibing Chen.