Applied/GPS/Spring2012

GPS Applied Mathematics Seminar

The GPS (Graduate Participation Seminar) is a weekly seminar by and for graduate students. If you're interested in presenting a topic or your own research, contact the organizers: Sarah Tumasz, Li Wang, and Zhennan Zhou.

All seminars are on Mondays from 2:25 to 3:15 in B235 Van Vleck. Speakers should aim for their talk to last no longer than 45 minutes.

Spring 2012

date	speaker	title
Feb 6	Zhan Wang	Hydroelastic solitary wave and its application in ice problem
Feb 13	Sarah Tumasz	What is Topological Mixing?
Feb 20	Zhennan Zhou	Semi-classical analysis of Schrodinger equation with periodic potential
Feb 27	Li Wang	A Jin-Xin-Glimm scheme for hyperbolic conservation laws
Mar 5
Mar 12
Mar 19	Xiaoqian Xu	Interpolation of Linear Operators
Mar 26	Yu Sun	Matching-pursuit/split-operator-Fourier-transform simulations of quantum dynamics
Apr 2	Spring Break
Apr 9	Lei Li	Landau–Zener formula
Apr 16	Leland Jefferis	Wigner Decomposition Methods for High Frequency Symmetric Hyperbolic Systems
Apr 23	Tete Boonkasame	A stable nonhydrostatic shallow-water model for two-layer flows
Apr 30	Antonio Ache	Asymptotics for solutions of parabolic systems
May 7

Abstracts

Monday, February 6: Zhan Wang

Hydroelastic solitary wave and its application in ice problem

The study of deformation of a floating ice sheet has applications in polar regions where ice cover is used for roads or runways and there is an interesting on the safe use of these transport links. We use the full potential model to study the forced and unforced dynamics of hydroelastic waves near the minimum phase speed in deep water. This is a joint work with Paul Milewski and J.-M. Vanden-Broeck.

Monday, February 13: Sarah Tumasz

What is Topological Mixing?

In this talk, I hope to provide an answer to the question 'What is Topological Mixing?' This will be a very introductory level talk, and I'll attempt to give intuitive, rather than technical, definitions. I'll discuss the basics of mixing, topology, and braids (as they apply) and then give some examples of applications.

Monday, February 20: Zhennan Zhou

Semi-classical analysis of Schrodinger equation with periodic potential

abstract In this talk, I plan to (at least try to) convince you that for Schrodinger equations, we need more analytic insight to build up reliable numerical schemes. I will introduce basic asymptotic methods for semiclassical limits, and explain why the situation changes dramatically for highly oscillatory potentials.

Monday, February 27: Li Wang

A Jin-Xin-Glimm scheme for hyperbolic conservation laws

We present a class of numerical scheme(called Jin-Xin-Glimm scheme) for scalar conservation law, which will be extended to hyperbolic system later. This scheme contains the advantages of Jin-Xin relaxation scheme, which is free of Riemann solver, and Glimm scheme, which is a sharp shock capturing method. This is a joint work with Frederic Coquel, Shi Jin and Jian-guo Liu.

Monday, March 19: Xiaoqian Xu

Interpolation of Linear Operators

I'll try to introduce the definition of linear operators of strong type(p,q), weak type(p,q) and restricted weak type (p,q). I'll also give the definition of weak Lp spaces and general Lorentz spaces. In my view, there is no relationship between this and applied math.

Monday, April 9: Lei Li

Landau-Zerner formula

Landau-Zerner formula describes the transition rate for a 2-level quantum mechanical system. It has many applications in various models. In BO approximation, it's the basis for the surface-hopping algorithm. I'll briefly talk about how to derive this formula following the original idea of Zener proposed in 1930s.

Monday, April 16: Leland Jefferis

Wigner Decomposition Methods for High Frequency Symmetric Hyperbolic Systems

We develop a method for computing physical observable for general symmetric hyperbolic systems using the high frequency limit of the Wigner Transform. Symmetric hyperbolic systems can represent many physically relevant systems of partial differential equations (PDE) such as Maxwell's equations, the elastic equations and the acoustic equations. Using the Wigner transform to analyze high frequency behavior has been done in special cases, but here we extend its application to the fully genearl case.

Monday, April 23: Tete Boonkasame

A stable nonhydrostatic shallow-water model for two-layer flows

The system of equations describing nonlinear and weakly dispersive internal waves between two layers of inviscid fluids bounded by top and bottom walls is known to be mathematically ill-posed despite the fact that physically stable internal waves have been observed. We obtain a stable nonhydrostatic model for this system and illustrate our results with solitary waves.

Monday, April 30: Antonio Ache

Asymptotics for solutions of parabolic systems

We will start by discussing the statement and some key of the key ideas involved in a result proved by Leon Simon in 1983 about the solution of certain elliptic variational problems by following the curves of "steepest descent". More importantly we will discuss applications to 3 problems

(1) The unique asymptotic limit problem for gradient flows, (2) The uniqueness of the cone structure at infinity for Ricci-flat manifolds, (3) The unique tangent problem for harmonic maps.

Some of the central aspects of the theory include the Lojasiewicz-Simon inequality, monotonicity formulas and the Liapunov-Schmidt reduction.