This is a joint work with Lina Chen and Prof. Xiaochun Rong. The Milnor Problem (modified) in the theory of group growth asks whether any finite presented group of vanishing algebraic entropy has at most polynomial growth. It is conjecture by Grigorchuk and Park to be true. We show that a positive answer to the modified Milnor Problem is equivalent to the Nilpotency Conjecture in Riemannian geometry: given n, d>0, there exists a constant /epsilon(n,d)>0 such that if a compact Riemannian n-manifold M satisfies that Ricci curvature >=-(n-1), diameter <=d and volume entropy </epsilon(n,d), then the fundamental group of M has a nilpotent subgroup of finite index. It is our hope that this equivalence will bring geometric tools into the study of Milnor Problem, since by the equivalence progresses made in either problem will shed a light on the other.

Let us consider a sequence of pointed n-manifolds with uniform Ricci curvature lower bound and uniform volume lower bound. By Gromov's pre-compactness theorem, up to a subsequence would converge in Gromov-Hausdorff sense to a metric space (X, d, p). In this talk we will consider the structure of such metric space X and some applications. We will first introduce the results of Cheeger-Colding, and then discuss our recent improvement. This is based on a joint work with Professors Jeff Cheeger and Aaron Naber.

A manifold is said to be of finite topological type if it is homeomorphic to the interior of a compact manifold with boundary. In this talk, I will give a brief introduction to the main results of complete manifolds with nonnegative Ricci curvature and quadratically nonnegatively curved infinity about the finite topological type. This includes some finiteness results under certain conditions of diameter growth (resp. volume growth) and some couterexamples of infinite topology with positive Ricci curvature. This is a joint work with Huihong Zhang.

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