Analysis and Its Applications Seminar

The Analysis and Its Applications seminar is organized by the "Analysis and Its Applications" research group. The seminar provides a forum in which faculty and graduate students present their work and exchange ideas on a broad range of topics, including singular limits of variational problems, eigenvalues of random matrices and graphs, spectral properties of elliptic operators, stability and instability of coherent structures and pattern formation.

Graduate students interested in applied analysis are particularly encouraged to attend.

The Analysis & Its Applications Seminar is held on Tuesdays at 12:30 PM, in Mathematics 402.

Spring 2008

January

01/22

Organizational meeting

01/29
Efi Efrati
Hebrew University of Jerusalem

Elastic Theory of Non-Euclidean Plates

Thin elastic sheets are very common in both natural and man-made structures. The configurations these structures assume in space are often very complex and may contain many length scales, even in the case of unconstrained thin sheets. We will show evidence of the simplicity of the intrinsic geometry leading to these complex three-dimensional configurations, and discuss the mechanism of shaping thin elastic sheets through the prescription of intrinsic metric.

Current reduced (two-dimensional) elastic theories devised to describe thin structures treat either plates (flat bodies having no structure along their thin dimension) or shells (non-flat bodies having a non-trivial structure along their thin dimension). We propose the concept of non-Euclidean plates, which are neither plates nor shells, to approximate many naturally formed thin elastic structures. We derive a thin plate theory which is a generalization of existing linear plate theories for large displacements but small strains, and arbitrary intrinsic geometry. We conclude by surveying some experimental results for laboratory-engineered non-Euclidean plates.

February

02/05
Ken McLaughlin
Department of Mathematics
The University of Arizona

A few open directions for research in random matrices, integrable PDEs, and Riemann-Hilbert problems

I will describe some open questions in the asymptotic analysis of Riemann-Hilbert problems, random matrices, and integrable PDEs.

02/12
Lotfi Hermi
Department of Mathematics
The University of Arizona

A Class of New Inequalities for the Eigenvalues of the Dirichlet Laplacian

For the eigenvalues of a Schroedinger operator with a nonnegative potential, Dolbeault, Felmer, Loss, and Paturel proved (JFA, 2006) a class of inequalities of Lieb-Thirring type relating functions of the spectrum to (appropriate) integrals which depend on the potential. These inequalities include in particular the classical Golden-Thompson (also called Kac-Ray) inequality and a gamut of new inequalities for the spectral zeta function and other functions.

In this work, we use the Bethe sum rule and transform methods to prove a parallel set of inequalities for the eigenvalues of the fixed membrane problem. We also prove the equivalence of the Berezin-Li-Yau inequality with a classical inequality of Kac, analyze the work of A. Melas in a new light, and propose new conjectures.

This is joint work with Professor Evans Harrell of Georgia Tech.

A preprint is available at arxiv.org/abs/0712.4088

02/26
Karl Glasner
Department of Mathematics
The University of Arizona

Migration Phenomena of Liquid Film Droplets and Surprises about Singularly Perturbed Gradient Flows

Gradient flows ostensibly have dynamics that (1) result from the decrease of energy and (2) do not exhibit oscillations. Neither of these statements is true for limits of singularly perturbed systems, however. This talk illustrates these points in the context of liquid film evolution.

Energy-driven coarsening processes arise as the late-stage dynamics of many problems. Two examples are the spinodal decomposition of binary mixtures and the dewetting of an unstable film of viscous liquid. The first case gives rise to Ostwald ripening, where large particles grow at the expense of smaller ones by exchanging material. Migration of the particles as a result of the ambient mass flux is a slower process and is usually ignored. In contrast, the nearly singular kinetics associated with the hydrodynamics of liquid films makes the role of migration significant. We discuss this phenomenon from both a variational and a perturbation theory point of view.

The effects of gravity or hoop stress can also lead to migration, but for different reasons. Interaction of droplets or fluid ridges is shown to give rise to a system which produces neighbor-neighbor repulsion and oscillatory dynamics.

March

03/04
Nick Ercolani
Department of Mathematics
The University of Arizona

Applications of Concentration Inequalities

Concentration inequalities provide effective tools in probabilistic contexts to deal with deviations of random variables from their expectations. We will give an introduction to concentration inequalities in a context that has applications, or potential applications, to questions in random matrix theory.

03/11
Robert Jenkins
Program in Applied Mathematics
The University of Arizona

An Asymptotic Method for Riemann-Hilbert Problems

In this talk I hope to explain the Deift-Zhou steepest descent method for oscillatory Riemann-Hilbert problems (RHPs), which can be thought of as a method analogous to the steepest descent method for obtaining asymptotic limits of integral expressions. The asymptotic limits I will consider come from RHPs associated with the nonlinear Schrodinger equation.

03/25
Joceline Lega
Department of Mathematics
The University of Arizona

Collective Behaviors, Lyapunov Modes, and Random Matrices

I will start with examples of models that describe collective behaviors in animal herds, fish schools, and bird flocks, and explain how some of these ideas can be extended to describe the dynamics of bacterial systems which do not communicate with one another. This will take us to problems related to the dynamics of gases, such as for instance granular gases. In that context, I will show simulations that explore the question of how the macroscopic behavior of a system of interacting particles can be affected by changing the collision rules between these particles. I will then explain how I think this could be related to the Lyapunov modes of the dynamics of the system. The structure of these modes has been widely studied in the literature and, in particular, the modes associated with small (in absolute value) Lyapunov exponents are known to have large-scale correlations. I will summarize the contents of a paper by Eckmann and Gat (J. Stat. Phys., 2000), which tries to explain this phenomenon by looking at the eigenvectors of a random matrix.

April

04/01
Sunhi Choi
Department of Mathematics
The University of Arizona

Free Boundary Regularity for the Stefan Problem

I will talk about the regularity properties of the free boundary for the Stefan problem, which models the phase transition between solids and liquids, and will present a recent work: If the initial free boundary is Lipschitz with a small Lipschitz constant, then the weak (viscosity) solution of the one-phase Stefan problem immediately regularizes and is smooth in space and time, for a small positive time.

04/08
Ibrahim Fatkullin
Department of Mathematics
The University of Arizona

Diffusive Transport in Nematics

One of the commonly used equations describing nematic liquid crystals is the so-called Doi-Smoluchowski equation. In essence, it is a kinetic equation for evolution of the orientation probability density of the system. I will present an analogue of this equation for spatially inhomogeneous systems and will discuss the associated problems of moment closure and reduction to Ginzburg-Landau type dynamics.

04/15
Dorin Dumitrascu
Department of Mathematics
University of Arizona

Groupoid Actions on C*-algebras

The main goal of the talk is to discuss the notion of groupoid action on a C*-algebra and point out some of the difficulties that one has to deal with. I will use two examples to show the usefulness of groupoids. The first is the characterization of an AF algebra as the C*-algebra of a principal groupoid, and the second is the use of the tangent groupoid in index theory.

04/22
Dorin Dumitrascu
Department of Mathematics
The University of Arizona

Groupoid Actions on C*-algebras, Part II

After reviewing some more examples of topological groupoids, I will present the notion of groupoid action on a topological space. This can then be generalized to groupoid actions on C*-algebras. I will end up with a discussion of the tangent groupoid, its associated C*-algebra, and their use in index theory.

04/29
Silvia Madrid-Jaramillo
Program in Applied Mathematics
The University of Arizona

Numerical computation of the Evans function with application to the stability of small deformations of an elastic rod

The Evans function is an analytic function that can be used to investigate the spectral stability of one-dimensional localized solutions to nonlinear PDEs. It is defined in such a way that its zeros on the right complex plane are eigenvalues of the corresponding linearized operator. The Evans function is a Wronskian-like function and can in principle be evaluated by a shooting method. In this situation, however, numerically evaluated solutions corresponding to linearly independent initial conditions tend to become linearly dependent, which may lead to spurious zeros of the Evans function. I will argue that for localized solutions that are sufficiently narrow, it is possible to accurately compute the Evans function using a shooting method.

As an example, I will consider the spectral stability of a two-parameter family of traveling wave solutions to two coupled nonlinear Klein-Gordon equations. These are envelope equations that model the dynamics of small deformations of an elastic rod near a writhing bifurcation. Numerical results are in agreement with analytical results obtained by S. Lafortune and J. Lega.