Let $M$ be a compact Riemannian manifold on which a compact Lie group $G$ acts by isometries. In this talk I will explain how the symmetry induces extra structures in the spectrum of Laplace-type operator, and how to apply symplectic techniques to study the induced equivariant spectrum. This is based on joint works with V. Guillemin and with Y. Qin.

Let M be a closed Riemannian manifold on which the integral ofthe scalar curvature is nonnegative. Suppose a is a symmetric (0,2) tensor field whose dual (1,1) tensor A has n distinct eigenvalues, and tr(A^k) are constants for k = 1, ..., n-1. We show that all the eigenvalues of A are constants,generalizing a theorem of de Almeida and Brito in 1990 to higher dimensions.As a consequence, a closed hypersurface M in S^{n+1} is isoparametric if one takes a above to be the second fundamental form, giving affirmative evidence to Chern's conjecture. This is a joint work with Zizhou Tang and Dongyi Wei.

We study the dynamics of a delayed diffusive hematopoiesis model with two types of Dirichlet boundary conditions. For the model with a zero Dirichlet boundary condition, we establish global stability of the trivial equilibrium under certain conditions, and use the phase plane method to prove the existence and uniqueness of a positive spatially heterogeneous steady state. We further obtain delay-independent as well as delay dependent conditions for the local stability of this steady state. For the model with a non-zero Dirichlet boundary condition, we show that the only positive steady state is a constant solution. Results for the local stability of the constant solution are also provided. By using the delay as a bifurcation parameter, we show that the model has infinite number of Hopf bifurcation values and the global Hopf branches bifurcated from these values are unbounded, which indicates the global existence of periodic solutions.

We study a general viral infection model with spatial diffusion in virus and two types of infection mechanisms: cell-free and cell-to-cell transmissions. The model is a partially degenerate reaction-diffusion system, whose solution map is not compact. We identify the basic reproduction number and explore its properties when the virus diffusion parameter varies from zero to infinity. Moreover, we demonstrate that the basic reproduction number is a threshold parameter for the global dynamics of our model system: the infection and virus will be cleared out if the basic reproduction number is no more than one. On the other hand

We propose a novel approach for variable screening in sparse multivariate additive models with random effects by use of null-beamforming on the data. The new approach includes two stages. In Stage 1, we approximate each nonparametric component by a linear combination of spline basis functions. Consequently, we convert the above problem to that of selecting groups of coefficients in a multivariate regression model with vector-valued covariates. In Stage 2, we conduct a series of filtering operations (called beamforming) by projections of the multiple response observations into each covariate space; each filter is tailored to a particular covariate and resistant to interferences originating from other covariates and from background noises

Graphical models are commonly used in representing conditional independence between random variables, and learning the conditional independence structure from data has attracted much attention in recent years. However, almost all commonly used graph learning methods rely on the assumption that the observations share the same mean vector. In this paper, we extend the Gaussian graphical model to the setting where the observations are connected by a network and propose a model that allows the mean vectors for different observations to be different. We have developed an efficient estimation method for the model and demonstrated the effectiveness of the proposed method using simulation studies. Further, we prove that under the assumption of "network cohesion", the proposed method can estimate both the inverse covariance matrix and the corresponding graph structure accurately.

We develop a modified Poisson-Nernst-Planck model to include Coulomb many-body properties in electrolytes, which also takes the ion-size effect into account and is expected to provide more accurate prediction for ion dynamics with microscopic confinement. Asymptotic expansions are performed to remove the multiscale properties of the equations and also used to understand dielectric properties near interfaces. Furthermore, we discuss numerical strategies to solve the resulted PDEs and show numerical results to demonstrate the performance of our numerical methods.

We study Bayesian inference methods for solving linear inverse problems, focusing on hierarchical formulations where the prior or the likelihood function depend on unspecified hyperparameters. In practice, these hyperparameters are often determined via an empirical Bayesian method that maximizes the marginal likelihood function, i.e., the probability density of the data conditional on the hyperparameters. Evaluating the marginal likelihood, however, is computationally challenging for large- scale problems. In this work, we present a method to approximately evaluate marginal likelihood functions, based on a low-rank approximation of the update from the prior covariance to the posterior covariance. We show that this approximation is optimal in a minimax sense. Moreover, we provide an efficient algorithm to implement the proposed method