Selected math problems in modern physics
Physics DepartmentUniversity of Nevada, Reno
Tuesday, November 30, 2:30-3:45pm, DMS 104
Abstract: Day-to-day work of a theoretical physicist involves math. In this talk, I will set up a physics background and highlight several problems where our group could benefit from expertise of a professional mathematician. In particular, I will talk about improving precision timekeeping using entangled states and challenge the audience to find a useful measure of entanglement in this context. I will also discuss atomic parity violation and outline challenges in many-body methods required for precision interpretation of experiments. Familiarity with basic concepts of quantum mechanics will be assumed.
Three-dimensional finite element models of earthen levee seepage and stability
Coastal and Hydraulics Laboratory, U.S. Army Corps of Engineers
Thursday, October 21, 2:30-3:45pm, DMS 103
Abstract: Levee performance under flood conditions can be determined by a wide variety of processes, but two of the most significant are flow and deformation in the porous material that constitutes the levee. Flow in levees is modeled as variably saturated flow in porous media, either as a free boundary problem for the saturated subdomain of the levee, or using the Richards equation for variably saturated flow in the entire levee. Levee deformation can be modeled using elastic-plastic constitutive theory such as the classical Mohr-Coulomb theory. The two processes are coupled because the seepage exerts a body force on the soil skeleton, and because deformation of the soil skeleton modifies important hydraulic properties. The levee is stable if it can maintain an equilibrium configuration under flood conditions; otherwise, the levee slope fails by undergoing large deformations. In this presentation I will present three-dimensional finite element simulations of several test problems for levee seepage and stability, including some real-world application problems. Our current approach uses standard continuous, piecewise polynomial finite element spaces on tetrahedral meshes. Due to the range of scales involved, the simulations must be carried out on massively parallel architectures. We use linear elements for flow and quadratic elements for deformation, but the character of the underlying PDE and solutions makes it very appealing to consider the prospect of solution-based adaption and mixed finite elements.
Topological Insulators and Joint Approximate Diagonalization of Matrices
Department of Mathematics and Statistics, University of New Mexico
Thursday, October 14, 2:30-3:45pm, DMS 104
Abstract: Only commuting families of self-adjoint matrices can be jointly diagonalized, but that does not stop physicists and electrical engineers from trying. Settling for a unitary change of basis that brings all the matrices "close'' to diagonal is often good enough. In signal processing, joint approximate diagonalization has led to effective algorithms for blind source separation. In quantum chemistry, joint approximate diagonalization can find a basis of low-energy space consisting of localized electron states.
There are limits to joint approximate diagonalization, some of which can be explained by $K$-theory. The choice of scalar field is surprisingly important. Theorems regarding almost commuting unitary matrices cannot be translated properly to results on almost commuting orthogonal matrices unless one takes care with the distinction between $KO$ and $KU$ groups.
Much of the talk will be on joint work with Matt Hastings on topological insulators. These are intriguing states of matter where a sample is conducting on the boundary, yet insulating in the interior. Numerical models of topological insulators suggest that fast algorithms to distinguish topological insulators from ordinary insulators can be found based on the $K$-theory obstructions to joint approximate diagonalization.
On Markov Chain Monte Carlo (MCMC) and Mixing Rates
Yevgeniy Kovchegov, Assistant Professor
Department of Mathematics, Oregon State University
Thursday, September 16, 2:30-3:45pm, DMS 104
Abstract: Markov Chain Monte Carlo (MCMC) is a method to simulate a desired probability distribution via constructing a Markov chain whose stationary distribution is the one we need. Mixing time describes the rate of convergence of a Markov chain to its stationary distribution.
We will give examples of Gibbs sampling algorithms (also known as Glauber dynamics). We will explain how strong stationary time and coupling are used to obtain bounds on mixing time. We will also discuss new approaches to coupling method and their applications.