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I will discuss some results on the classification of convex compact ancient solution to the generalized curve shortening flow (the speed of evolution is the curvature of the curve to some power). In particular, I will show that the only convex compact ancient solution to the affine curve shortening flow (also known as fundamental equation of image processing) is a shrinking ellipse.

An extension of the Lutwak projection inequality is established in the framework of Orlicz Brunn -Minkowski theory. This new result directly yields the Orlicz Brunn-Minkowski inequality for intrinsic volumes.

In this talk, I will first give an introduction on Tian's properness conjecture concerning on an analytic charactarization of the existence of canonical metrics in Kahler geometry. Then I will focus on compactifications of reductive Lie groups. The main results are criterion theorems of the properness of two important functionals——Ding functional and Mabuchi's K-energy on these manifolds. In particular, the existence of Kahler-Einstein metrics, Kahler-Ricci solitons and Mabuchi's generalized Kahler-Einstein metrics on Fano compactifications of reductive Lie groups can be established.

eaction-diffusion epidemic models are proposed in order to understand how spatial heterogeneity influences the transmission dynamics of infectious diseases. In this talk, I will first present our recent numerical studies of the seasonal influenza epidemics in Puerto Rico based on a diffusive susceptible-infected-recovered model. Our simulations demonstrate that even simple diffusive epidemic models have potential applications in real-world situations. Then I will present our theoretical investigations of a reaction-diffusion vector-host epidemic model.

By the super version of the Wedderburn Theorem, a finite dimensional complex simple (associative) superalgebras is isomorphic to either a (full) matrix superalgebra or a queer matrix superalgebra. This gives rise to two series Lie superalgebras: the general linear Lie superalgebra and the queer Lie superalgebra. In this talk, I am going to introduce the queer q-Schur superalgebras associated with the latter and discuss their fundamental structure. The Schur-Weyl-Sergeev duality involving the quantum queer supergroup and Hecke-Clifford superalgebra will also discussed.

Vector-borne infectious diseases may involve horizontal transmission between hosts in addition to transmission from infected vectors to susceptible hosts. In this talk, we develop a vector-borne disease model with general nonlinear incidence rates and continuous age structures in both infectious hosts and vectors. We first study the existence and local stability of steady states, which is completely determined by the basic reproduction number. With the assistance of the existence of a global attractor and the uniform persistence, a threshold dynamics is established by employing the Fluctuation Lemma and the approach of Lyapunov functionals.

In this talk, we discuss the Diederich-Fornæss index in several complex variables. A domain Ω in C^n is said to be pseudoconvex if -log(-δ(z)) is plurisubharmonic in Ω, where δ is a signed distance function of Ω. The Diederich-Fornæss index has been introduced since 1977 as an index to refine the notion of pseudoconvexity. After a brief review of pseudoconvexity, we discuss this index from the point of view of geometric analysis. We will find an equivalent index associated to the boundary of domains and with it, we are able to obtain accurate values of the Diederich-Fornæss index for many types of domains.

This report considers methods for the solution of the time dependent Stokes Darcy problem that can be implemented by use of existing surface water and ground water codes. Porous media problems for groundwater flows overwhelmingly use mixed discretizations due to their superior conservation properties and the conservation for problems on large domains with relatively coarse meshes.