The classical de Rham-Hodge theory implies that each cohomology
class of a compact manifold is uniquely represented by a harmonic
form, signifying the important role of Laplacian in geometry. The talk aims
to explain some results relating curvature to the spectrum of Laplacian. We
plan to start by a brief overview for the case of bounded Euclidean domains
and compact manifolds, highlighting some of the fundamental contributions by
Peter Li and others. We then shift our focus to the case of complete
Riemannian manifolds. In particular, it includes our recent joint work with
Ovidiu Munteanu concerning the bottom spectrum of 3-manifolds with scalar
curvature bounded below.

There are many cooperative games which can be won with higher probability if the players are able to access quantum resources. In fact for some games, the players can have very small probability of winning with classical strategies, but the game can be won with probability one with quantum assistance. 

The theory of these games has recently been used to solve the Connes Embedding Problem, which had been open since the 1970's, and has been used to show that the mathematical models for describing quantum correlations are all different.

In this talk we introduce these ideas and focus on the family of synchronous games. For synchronous games there is an algebra whose representation theory determines whether or not they can be won with probability one.

This talk will be accessible to anyone with a basic knowledge of operators on a Hilbert space. 

The work reported here has been motivated by the design of lab-grown organs, such as a bioartificial pancreas. The design of lab-grown organs relies on using biocompatible materials, typically poroelastic hydrogels, to generate scaffolds to support seeded cells of different organs.  Additionally, to prevent the patient's own immune cells from attacking the transplanted organ, the hydrogel containing seeded cells is encapsulated between two semi-permeable, nano-pore size membranes/plates and connected to the patient's vascular system via a tube (anastomosis graft). The semi-permeable membranes are designed to prevent the patient's own immune cells from attacking the transplant, while permitting oxygen and nutrients carrying blood plasma (Newtonian fluid) to reach the cells for long-term cell viability.  A key challenge is to design a hydrogel with ``roadways'' for blood plasma to carry oxygen and nutrients to the transplanted cells.

We present a complex, multi-scale model, and a first well-posedness result in the area of fluid-poroelastic structure interaction (FPSI) with multi-layered structures modeling organ encapsulation. We show global existence of a weak solution to a FPSI problem between the flow of an incompressible, viscous fluid, modeled by the time-dependent Stokes equations, and a multi-layered poroelastic medium consisting of a thin poroelastic plate and a thick poroelastic medium modeled by a Biot model. Numerical simulations of the underlying problem showing optimal design of a bioartificial pancreas, will be presented. This is a joint work with bioengineer Shuvo Roy (UCSF), and mathematicians Yifan Wang (UCI), Lorena Bociu (NCSU), Boris Muha (University of Zagreb), and Justin Webster (University of Maryland, Baltimore County). 

A random nxm matrix gives a random linear transformation from \Z^m to \Z^n (between vectors with integral coordinates).  Asking for the probability that such a map is injective is a question of the non-vanishing of determinants.  In this talk, we discuss the probability that such a map is surjective, which is a more subtle integral question.  We show that when m=n+u, for u at least 1, as n goes to infinity, the surjectivity probability is a non-zero product of inverse values of the Riemann zeta function.  This probability is universal, i.e. we prove that it does not depend on the distribution from which you choose independent entries of the matrix, and this probability also arises in the Cohen-Lenstra heuristics predicting the distribution of class groups of real quadratic fields.  This talk is on joint work with Hoi Nguyen.

In this talk, I will first give an introduction to some models and algorithms from two different fields: (1) machine learning, including logistic regression, support vector machine and deep neural networks, and (2) numerical PDEs, including finite element and multigrid methods.  I will then explore mathematical relationships between these models and algorithms and demonstrate how such relationships can be used to understand, study and improve the model structures, mathematical properties and relevant training algorithms for deep neural networks. In particular, I will demonstrate how a new convolutional neural network known as MgNet, can be derived by making very minor modifications of a classic geometric multigrid method for the Poisson equation and then explore the theoretical and practical potentials of MgNet.

This is a joint talk of Applied and Compuational Math Seminar.