Fri, Apr 4

Applied Math Seminar
Tamas Horvath (applied math, Oakland U)
Space-Time Finite Element Methods - The Good, the Bad and the Ugly
3:00PM, Math 250

Partial differential equations posed on moving domains arise in many applications such as air turbine modeling, flow past airplane wings, etc. The time-dependent nature of the flow domain poses an additional challenge when devising numerical methods for the discretization of such problems. One alternative when dealing with time-dependent domains is to pose the problem on a space-time domain and apply, for example, a finite element method in both space and time. These space-time methods can easily handle the time-dependent nature of the domain. In this talk, we present a space-time hybridizable discontinuous Galerkin method for the discretization of the incompressible Navier-Stokes equations on moving domains. This discretization is pointwise mass conserving and pressure robust, even on time-dependent domains. Moreover, high order can be achieved both in space and time. Numerical experiments will demonstrate the capabilities of the method.

Wed, Apr 9

Analysis Seminar
Hanfeng Li, SUNY at Buffalo
Local entropy theory, combinatorics, and local theory of Banach spaces
4:00PM, 250 Math Building

In 1995 Glasner and Weiss showed that if a continuous action of a countably infinite amenable group on a compact metrizable space X has zero entropy, then so does the induced action on the space of Borel probability measures on X. I will discuss a strengthening of the Glasner-Weiss result, in the framework of local entropy theory,  based on a new combinatorial lemma. I will also present an application of the combinatorial lemma to the local theory of Banach spaces. This is joint work with Kairan Liu.

Wed, Apr 16

Analysis Seminar
Ryo Toyota, Texas A&M University
Expanders, geometric property (T), and warped cones
4:00PM, 250 Math Building

Warped cones are metric spaces associated with dynamical systems, where their large-scale geometric properties reflect the dynamical properties of the underlying actions. In this talk, we discuss a large-scale invariant, called geometric property (T), for warped cones and show that if an action is ergodic, free, measure preserving and isometric on a Riemannian manifold, then the associated warped cone does not possess geometric property (T). This result negatively answers an open problem: whether the warped cone of an ergodic action by a group with property (T) possesses geometric property (T), and gives new examples of (super-)expanders without geometric property (T). This is based on a joint work with Jintao Deng.

Fri, Apr 18

Applied Math Seminar
Abner Salgado (University of Tennessee, Knoxville)
Energy, pointwise, and free boundary approximation of the obstacle problem for nonlocal operators

3:00PM, Math 250

We consider the obstacle problem for a nonlocal elliptic operator, like the integral fractional Laplacian of order 0<s<1. We derive regularity results in weighted Sobolev spaces, where the weight is a power of the distance to the boundary. These are then used to obtain optimal error estimates in the energy norm. 

Next, we develop a monotone, two-scale discretization of the operator, and apply it to develop numerical schemes. We derive pointwise convergence rates for linear and obstacle problems governed by such operators. As applications of the monotonicity, we provide error estimates for free boundaries and a convergent numerical scheme for a concave fully nonlinear, nonlocal, problem.

This presentation is based on several works in collaboration with: A. Bonito, J.P. Borthagaray, W. Lei, R.H. Nochetto, and C. Torres.

Fri, Apr 25

Applied Math Seminar
Sathyanarayanan Chandramouli (University of Massachusetts Amherst)
TBD
3:00PM, Math 250

TBD

Mon, May 5

Algebra Seminar
Claudiu Raicu, University of Notre Dame
Polynomial functors and stable cohomology
4:00PM, 250 Math Building

The theory of polynomial representations of the general linear group goes back to the thesis of Issai Schur at the turn of the 20th century. Such representations include the tensor, symmetric, and exterior powers of a vector space, and have been completely classified in the work of Schur when the underlying field is the complex numbers. While there has been significant progress since the work of Schur, the story over a field of positive characteristic remains largely unknown. In my talk I will describe some novel stabilization results for sheaf cohomology, and explain their connection to the study of polynomial representations / functors. This is based on joint work with Keller VandeBogert.