Recently Eskin-Kontsevich-Moller-Zorich prove my conjecture that the sum of the top $k$ Lyapunov exponents is always greater or equal to the degree of any rank $k$ holomorphic subbundle(They generalize the original context from Teichmuller curves to any local system over a curve with non-expanding cusp monodromies). Furthermore, they conjecture that equality of the subgroup of its Lyapunov exponents and degrees is related to the monodromy group being a thin subgroup of its Zariski closure. I will introduce some backgrounds on those conjectures and some applications to Teichmuller dynamics and Calabi-Yau type families.

In this talk, we study the volume of locally conformally hpyersphere in 5-dimensional Riemnnian manifold with sectional curvature between 0 and 1. In particular, we obtain a sharp lower bound of the minimal locally conformally flat 4-hypersphere, which partially answer a question proposed by Mazet and Rosenberg. This is a joint work with Cui Qing.

A change point refers to a location or time at which observations or data obey two different models: before and after. These studies of change-point problems have found applications in a wide range of areas, including quality control, finance, environmetrics, medicine, genetics and geography. We propose a procedure for detecting multiple change-points in a mean-shift model. We first convert the change-point problem into a variable selection problem by partitioning the data sequence into several segments. Then, we apply a modified variance inflation factor regression algorithm to each segment in sequential order.

We study the Poisson-Nernst-Planck (PNP) system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges. Assuming point charges and that the Debye length is small relative to the domain size, we derive an asymptotic formula for the steady-state solution by matching outer and boundary layer solutions. The case of two ionic species has been extensively studied, the uniqueness of the solution has been proved, and an explicit expression for the solution has been obtained. However, the case of three or more ions has received significantly less attention.

Using the marked empirical processes, this paper develops a test of parametric specification in a nonlinear cointegrating regression model. Unlike the kernel-smoothed $U$-statistic considered in Gao, et al. (2009) and Wang and Phillips (2012), our new test statistic avoids the use of bandwidth, which has some advantages for practitioners. Simulations and a real data example show that our new test has a good finite sample performance. Another contribution of this paper is to provide a rigorous proof on weak convergence for a class of martingales, which is interesting in its own rights.

Stochastic multiscale modeling has become a popular approach to quantify uncertainty in multiscale models. The combination of multiscale features and complex uncertainty in models leads to great challenge to simulate the models and explore the propagation of uncertainty. To treat the difficulty, it is desirable to construct a sparse representation for the outputs of the multiscale models. In the talk, we present a sparse representation based on reduced multiscale basis methods and investigate uncertainty quantification for some porous media applications.

We consider the optimal dividend payment strategy for an insurance company, having two collaborating business lines, where their surplus processes are modelled by diffusion processes and the dividends paid by different business lines are weighted differently. We find the optimal dividend strategy when money is transferred between two business lines without transaction costs. We show the optimal value function is the unique continuous viscosity solution to the corresponding HJB equation when money is transferred with transaction costs. We also prove a verification theorem. Finally, we find the optimal solution to the problem with transaction costs in the symmetric case.

We consider the optimal investment and consumption decision of a CRRA investor who is facing capital gains tax and receives labor income. Assume that the investor cannot borrow against future labor income. We investigate how the coexistence of capital gains tax and uninsurable labor income affects the investment and consumption decision. This work is jointly with Yaoting Lei and Hong Liu