Analysis and Its Applications Seminar
Archive

Fall 2007

August

08/21

Organizational Meeting

08/28
Tien-Tsan Shieh
Department of Mathematics
University of Arizona

The Onset Problem for a Thin Superconducting Loop in a Large Magnetic Field

We present a rigorous analysis of the eigenvalue problem associated with the onset of superconductivity for a thin domain in the presence of a large applied magnetic field. We prove the validity of Richardson and Rubinstein's formal result which reveals that in this double limit of thin domain and large field, the appropriate Rayleigh quotient differs from the standard one for order 0(1) applied fields through the addition of a potential depending on the field. This also demonstrates a parabolic background for the oscillatory phase transition curve between the normal and superconducting state.

September

09/04
Tien-Tsan Shieh
Department of Mathematics
University of Arizona

The Onset Problem for a Thin Superconducting Loop in a Large Magnetic Field (continued from 8/28)

We present a rigorous analysis of the eigenvalue problem associated with the onset of superconductivity for a thin domain in the presence of a large applied magnetic field. We prove the validity of Richardson and Rubinstein's formal result which reveals that in this double limit of thin domain and large field, the appropriate Rayleigh quotient differs from the standard one for order 0(1) applied fields through the addition of a potential depending on the field. This also demonstrates a parabolic background for the oscillatory phase transition curve between the normal and superconducting state.

09/11
Jorge Ramirez
Department of Mathematics
University of Arizona

Probabilistic Representations, Majorizing Kernels, and Spectral Behavior in Fluid Dynamics: The Case of Burgers Equation

Probabilistic representations of deterministic partial differential equations often provide important insight into the underlying physical processes, e.g. Brownian motion as a model for the motion of indivial particles in a solute whose concentration satisfies the advection-diffusion equation. Recently, a family of probabilistic representations for the solution to the Navier-Stokes equation (NSE) has been discovered. This family is parametrized by a collection of functions dubbed "majorizing kernels." For any such majorizing kernel h, the Fourier transform of the solution to NSE is written as the expected value of a random multiplicative process defined on a tree in Fourier space, where the contribution of each frequency to the solution is determined by h. Even though this structure irresistibly suggests a connection with the energy cascade model of turbulent flows, no firm grounds for a physical interpretation of the probabilistic representation exist. In particular, the construction of majorizing kernels for which the spectrum of the solution exhibits the scaling properties predicted by the turbulent cascade model, would be very desirable. In this talk I will show some elementary ongoing work on the formulation of this problem for Burgers equation: a simpler nonlinear model in fluid dynamics exhibiting analogous structure as NSE in its probabilistic representation and spectral properties.

09/18
Lotfi Hermi
Department of Mathematics
University of Arizona

Improved Universal Weyl-Type Bounds for Eigenvalues of the Dirichlet Laplacian

Trace identities of the type derived by Harrell-Stubbe, and later generalized by Levitin-Parnovski, proved to be a very efficient procedure to produce universal Yang-type bounds for eigenvalues of the Dirichlet Laplacian.

In this talk we show how these identities can be used to produce new universal Weyl-type bounds for averages of eigenvalues, and provide alternative routes to the Berezin-Li-Yau inequality as viewed by Laptev and Weidl.

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

A preprint is available at lanl.arxiv.org/abs/0705.3673

09/25

Organizational meeting for the New Trends in Nonlinear Analysis conference

<a href="math.arizona.edu/~ntna2007">math.arizona.edu/~ntna2007</a>

October

10/02
Leonard Friedlander
Department of Mathematics
University of Arizona

Spectrum of Narrow Infinite Waveguides

This is a report on joint work with Mikhael Solomyak of the Weizmann Institute. We study eigenvalue asymptotics of the Dirichlet Laplacian in narrow infinite planar domains as their thickness goes to 0. In the case of periodic domains, we derive estimates on the band widths.

10/09
Ken McLaughlin
Department of Mathematics
University of Arizona

Universality: Asymptotic Analysis of Riemann-Hilbert Problems via 'dbar' Methods

I will describe two asymptotic results: one concerning random matrix theory and one concerning the NLS equation. From the bowels of those considerations will emerge a crazy method for analyzing the asymptotic behavior of an integral. Then some open directions for research (aimed at graduate students) will be discussed.

10/16
Liana Dawson
Deparment of Mathematics
University of Arizona

Unique Continuation for Fifth-order Dispersive Equations

This talk will be concerning uniqueness properties of solutions to nonlinear dispersive equations. We will begin by reviewing some of the known existence and uniqueness results for dispersive equations. Then we will establish a unique continuation property for fifth-order equations. The goal will be to show that if the difference of two solutions to a fifth-order nonlinear dispersive equation decays sufficiently fast at infinity at two times, then the solutions are equal.

10/30
Ibrahim Fatkullin
Department of Mathematics
University of Arizona

Stochastic Evolution in Bistable Systems and Diffusion-Annihilation Processes

This talk is a logical continuation of my earlier lectures on asymptotic reductions of stochastically perturbed gradient flows. Whereas the SDE case may be analyzed rigorously there still exist many difficulties in the treatment of the SPDE case. I will illustrate some of the methods and ideas on the example of reduction of stochastically perturbed Allen-Cahn–type equations to diffusion-annihilation processes (system of interacting Brownian particles annihilating on collision).

November

11/06
Joe Watkins

The Wright-Fisher Diffusion Process and an Application to Queues and Bacterial Recombination

In this talk, I will develop, using a duality argument, an identity stating that the Laplace transform of the length of a contiguous bacterial recombination region equals the probability of choosing a given allele in a stationary population evolving according to the one-dimensional Wright-Fisher diffusion model. Beyond giving us an improved inferential strategy for parameter estimation in bacterial recombination, the matching of the selection and recombination parameters in the identity also suggests the existence of an intriguing connection between ancestral recombination graphs and ancestral selection graphs. This work is joint with Xavier Didelot of Warwick University and Jay Taylor of the University of Oxford.

11/13
Joceline Lega

Molecular Dynamics Simulations of Live Particles

I will show results of molecular dynamics simulations of hard disks with non-classical collision rules. In particular, I will focus on how local interactions at the microscopic level between these particles can lead to large-scale coherent dynamics at the mesoscopic level.

This work is inspired by collective behaviors, in the form of vortices and jets, recently observed in bacterial colonies. I will start with a brief summary of basic experimental facts and review a hydrodynamic model developed in collaboration with Thierry Passot (Observatoire de la Côte d'Azur, Nice, France). I will then motivate the need for a complementary approach that includes microscopic considerations, and describe the principal computational issues that arise in molecular dynamics simulations, as well as the standard ways to address them. Finally, I will discuss how classical collision rules that conserve energy and momentum may be modified to describe ensembles of live particles, and will show results of numerical simulations in which such rules have been implemented. Randomness, included in the form of random reorientation of the direction of motion of the particles, plays an important role in the type of collective behaviors that are observed.

I will explain the numerical and multi-scale aspects of this line of work, but also take advantage of this talk to discuss potential connections with dynamical systems and random matrices, with the hope that ideas for synergistic activities might emerge.

11/20
Dorin Dumitrascu
Department of Mathematics
The University of Arizona

Property T Groups and Non-Invertibility of C*-Algebra Extensions

In the theory of C*-algebras, the extensions play a fundamental role because they generate the cycles of bivariant K-theories (in particular those of K-homology). A group G is said to have property T of Kazhdan if every continuous affine action of G on a real Hilbert space has a fixed point. This “rigid” behavior is opposite in a sense to that of similar actions of amenable groups. The main goal of the talk is to discuss, following the 1991 Annals paper of Simon Wassermann, how property T can be used to provide examples of non-invertible extensions. The needed facts related to extensions and property T will be also covered.

December

12/04
Mauro Carfora
Department of Nuclear and Theoretical Physics
University of Pavia and Istituto Nazionale di Fisica Nucleare (INFN)

The Conjugate Linearized Ricci Flow

We first introduce the conjugate linearized Ricci flow, a version of the linearization of the Ricci flow motivated by Perelman’s treatment of Ricci flow as a gradient flow. The flow takes into account the diffeomorphism invariance of the Ricci flow, which is the only obstruction to its strict parabolicity. We characterize the conjugate linearized Ricci flow on closed three-manifolds of bounded geometry and discuss its properties. In particular, we express the evolution of the metric and of its Ricci tensor in terms of the backward heat kernel of the conjugate linearized Ricci flow. These results provide various conservation laws and monotonicity formulas for the linearized flow. These results may be of interest to both analytical treatment of Ricci flow and physical applications of Ricci flow.

12/18
Sergey A. Cherkis
Permanent Lecturer, School of Mathematics
Trinity College, Dublin

Bows and Quivers: Instantons on Gravitons

Yang-Mills Instantons are connections on vector bundles with self-dual curvature, while Self-Dual Gravitational Instantons are Riemannian manifolds with self-dual Riemann curvature. Both are examples of integrable systems. We consider a problem of finding Yang-Mills Instantons on Gravitational Instantons. The solution is formulated in terms of Bow diagrams, which encode the instanton data. As in the case of instantons on flat spaces, one can obtain explicit solutions via the Nahm transform, which is the nonlinear generalization of the Fourier transform.