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.

Given a random elliptic or hyperelliptic curve of genus g over Q, how many rational points do we expect the curve to have? Equivalently, how often do we expect a random polynomial of degree n to take a square value over the rational numbers?  In this talk, we give an overview of recent conjectures and theorems giving some answers and partial answers to this question.

The talk will consist of two loosely connected parts: set-theoretic and computer science. We give an overview (no technical details) of our results on Galois-Tukey connections as a general framework for problem reduction. Boolean structure of absolutely divergent series gives rise to several Boolean-like asymptotic structures. Second part deals with applications of many valued logic to preference modeling, querying top-k answers and learning each individual user preferences from behaviour data (especially we mention lack of real world benchmarks).
 

In the first half of this talk we will review several notions of coarse or weak
Ricci Curvature on metric measure spaces which include the work of Yann
Ollivier. The discussion of the notion of coarse Ricci curvature will serve as
motivation for developing a method to estimate the Ricci curvature of a an
embedded submaifold of Euclidean space from a point cloud which has applications
to the Manifold Learning Problem. Our method is based on combining the notion of
``Carre du Champ" introduced by Bakry-Emery with a result of Belkin and Niyogi
which shows that it is possible to recover the rough laplacian of embedded
submanifolds of the Euclidean space from point clouds. This is joint work with
Micah Warren.

Over the past two decades, Algebraic Multigrid (AMG) has become a prominent tool for the rapid solution of unstructured-grid problems. For nearly 20 years after its inception, AMG was largely ignored, due to the overhead and setup costs necessary to create the components of the algorithm. So long as geometrically structured grids were the norm, this condition persisted. In recent years, however, as unstructured grids became more common and as the problem sizes grew to demand large-scale massively parallel computers, AMG has emerged as a widely used methodology. In this talk we describe the basic components and theory of AMG. Here, the focus is on the underlying philosophy of the method. We then concentrate on the more recent advances in AMG, and in particular on the development of the Bootstrap AMG framework.