First-passage percolation is a model of a random metric on a infinite network. It deals with a collection of points which can be reached within a given time from a fixed starting point, when the network of roads is given, but the passage times of the road are random. It was introduced back in the 60's but most of its fundamental questions are still open. In this talk, we will overview some recent advances in this model focusing on the existence, fluctuation and geometry of its geodesics. Based on joint works with M. Damron and J. Hanson.

I will discuss some recent results on developing new factorizations
for matrices obtained from discretizing differential and integral
operators. A common ingredient of these new factorizations is the
interpolative decomposition for numerically low-rank matrices. As we
shall see, these factorizations offer efficient algorithms for
applying and inverting these operators. This is a joint work with
Kenneth Ho.

We propose a dynamically bi-orthogonal method (DyBO) to study time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the Karhunen-Loeve expansion minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. In this talk, we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the covariance matrix or to compute its eigen-decomposition. We further present an adaptive strategy to dynamically remove or add modes, perform a detailed complexity analysis, and discuss various generalizations of this
approach. Several numerical experiments will be provided to demonstrate the effectiveness of the DyBO method.

The general Ramanujan Conjectures for congruence subgroups of arithmetic groups, and approximations that have been proven towards them, are central to many diophantine applications. Recently analogous results have been established for quite general subgroups of GL(n,Z) called  "thin groups ". We will describe some of these and review some of their applications (mainly diophantine) as well as the ubiquity of thin groups.

In this talk we consider the discrete maximum principle (DMP) for the weak Galerkin (WG) discretization of a general anisotropic diffusion problem. Brief introduction to DMP and WG will be given. It turns out that the stiffness matrix of the discretization is not an M-matrix in general, and therefore the theory of  M-matrices, which has been commonly used for the study of DMP, cannot be applied. To avoid this difficulty, a reduced system is first obtained by eliminating certain degrees of freedom  and is shown to satisfy DMP under suitable mesh conditions. Then we establish DMP for the full weak Galerkin approximation.Numerical examples, including DMP-compatible mesh generation using BAMG, will be reported. This talk is based on a joint work with Dr. Weizhang Huang.