In this talk, we introduce some recent results on Heegner points and its application to arithmetic of elliptic curves.

We consider the existence of spatially inhomogeneous time-periodic orbit in a reaction-diffusion population model with spatial-temporal nonlocal delayed growth rate and Dirichlet boundary condition. When the dispersal kernel is of strong or weak type, the scalar reaction-diffusion equation with distributed delay is converted into a system of two or three reaction-diffusion equations without delay. We prove the existence of periodic orbits for the system which are equivalent to periodic orbits for the original scalar model.

Understanding the mechanisms driving predator-prey population dynamics and stability has been a central theme in the field of ecology. Although theoretical models developed over the past quarter century have demonstrated that predator-prey population dynamics can depend critically on age (stage) structure and duration and variability in development times of different life stages, unambiguous experimental support for this theory is nonexistent. We conducted an experiment with the cowpea weevil Callosobruchus maculatus, and its parasitoid Anisopteromalus calandrae,

Spreading speed theory provides a mathematical tool to analyze the demography and dispersal of invasive species. Based on biological records, the secondary spread of the European green crab, Carcinus maenas, has maintained a relatively consistent rate of advance for over 120 years covering a wide range of temperate latitudes and local hydrological environments along the Atlantic coast of North America. We analyzed presence/absence data for recently established green crab populations, empirically estimated the crab’s spread rate, and employed a discrete-time model to investigate the relationship between the spreading speed and demography and dispersal parameters. The model couples a matrix population model for population growth with integrodifference equations for dispersal.

HIV infection is still a serious health problem in the world. Effective combination therapy can control viral replication but cannot eradicate the virus. Mathematical models, combined with experimental data, have provided important insights into HIV dynamics and immune responses. In this talk, I will present some recent work on modeling HIV infection and treatment, such as HIV latency and persistence, viral blips, virus dynamics under different drugs, treatment intensification with additional drugs, and the slow time scale of target cell depletion. Model formulation, mathematical analysis, numerical simulation (deterministic or stochastic) and comparison with data will be presented. Implications of modeling results for viral control strategies will also be discussed.

In this talk I will introduce our recent work on a generalization of the Fenchel theorem to the 3-dimensional Lorentz space. The total curvature of a spacelike closed curve is the same as the length of its tangent indicatrix. The estimation of this length is non-trivial. Here we will explain our discovery of an alternative phenomenon about closed spacelike curves on the de-sitter space, which helps to establish the desired inequality. Moreover, this also helps to obtain a necessary and sufficient condition for a curve to be realizable as the tangent indicatrix.

It will be shown that many of the deep and beautiful results for minimal graphs in codimension 1 fail utterly in higher codimensions. More precisely, (1) For the case of dimension 2, the Dirichlet problem is solvable, but these solutions are not unique in general. (2) When the dimension is no less than 4, the Dirichlet problem is not even solvable. (3) The minimal graphs need not even be stable. (4) There exist Lipschitz solutions to minimal surface equations which is not smooth.

In this talk, I report progress my team has made on modeling of Ebola Virus Diseases (EVD), Middle-East Respiratory Syndrome (MERS) and global spatio-temporal pattern of human influenza. Firstly, I introduce a new approach to disentangle common and distinct components in the transmission rate of EVD across countries. Secondly, I compare the seasonality of MERS versus influenza in the Middle East. Finally, I report a skip-and-resurgence phenomenon of human influenza H1N1 in Europe/East Asia and the spatial pattern of proportions of Influenza B in the post-pandemic era.