## Colloquium 2013-2014

## April 25, Marco Merkli, Memorial University of Newfoundland

I will present the ideas of a proof that the spin-boson system has the property of "Return to Equilibrium", no matter how strongly the spin is coupled to the bosonic heat bath. The question of "Return to Equilibrium" of open quantum systems has been studied by many authors over the last years, but only for weakly coupled systems. Our approach is based on a polaron transformation (introduced by Leggett et al.) and spectral analysis of the generator of the dynamics. The common complex scaling method for the spectral analysis does not apply in the situation at hand. We use a variant of Mourre-theory to examine the spectrum.

This is joint work with Martin Koenenberg and Haifeng Song.

## April 18, Sevak Mkrtchyan, Carnegie-Mellon

The dimer model is the study of random perfect matchings on graphs, and has a long history in statistical mechanics. On the hexagonal lattice it is equivalent to tilings of the plane by lozenges and to 3D stepped surfaces called skew plane partitions - 3 dimensional analogues of Young diagrams with a partition removed from the corner. This particular instance of the model has been intensely studied in the past 15 years by Kenyon, Okounkov, Reshetikhin and many coauthors. I will discuss the scaling limit of the model under a certain family of measures called "volume"-measures, the limit-shape phenomenon in this model (a form of the law of large numbers), the effects of varying the boundary conditions on the limit shape, the nature of local fluctuations in various regions of the limit shape and connections with random matrix theory.

## April 9, Emmanuele DiBenedetto, Vanderbilt

The local positivity of solutions to logarithmically singular diffusion equations is investigated in some space-time domain $E\times(0,T]$. It is shown that if at some time level $t_o\in(0,T]$ and some point $x_o\in E$ the solution $u(\cdot,t_o)$ is not identically zero in a neighborhood of $x_o$, in a measure-theoretical sense, then it is strictly positive in a neighborhood of $(x_o,t_o)$. The precise form of this statement is by an intrinsic Harnack-type inequality, which also determines the size of such a neighborhood. Recent results on the space-analiticity of solutions will also be presented.

## Feb 20, Milivoje Lukic, Rice University

The spectrum of an operator is a generalization of the notion of the set of eigenvalues of a finite matrix. We are concerned with spectral properties of certain classes of operators, such as Schrodinger operators (central to quantum mechanics) and Jacobi matrices (tied to orthogonal polynomials). Work of Deift-Killip and Killip-Simon shows that L^2 perturbations of the free Schrodinger operator or free Jacobi matrix preserve a.c. (absolutely continuous) spectrum. This result is optimal on the L^p scale, so spectral properties of slower decaying perturbations can only be established under additional assumptions. In this talk, we will discuss several recent results on slowly decaying perturbations. Some of our results solve an open problem about a class of oscillatory decaying perturbations which includes almost periodic times decaying functions. In another approach, we describe the spectral consequences of L^2 bounded variation conditions. Finally, we discuss our recent contributions to higher-order Szego theorems; this includes the disproving of a conjecture of Simon and the first equivalence result in the regime of arbitrarily slow decay.

## Feb 3, Geng Chen, Georgia Tech

Abstract: It is well-known that solutions of nonlinear hyperbolic PDEs often develop singularities in finite time, such as cusps, and shock waves. The lack of regularity is the major difficulty in the construction of solutions, and in the justification of uniqueness if the solution exists. The singularities in the solutions generate rich phenomena which are often challenging to understand. In this talk, we discuss two important classes of hyperbolic PDEs, including compressible Euler equations, Camassa-Holm equation, and variational wave equations, modelling gas dynamics, shallow water and liquid crystal, respectively. We will show how the understanding of certain singular behaviors of the solutions helps to identify the difficulties, design appropriate methods, and lead to the resolution of several interesting open problems. This talk is based on a series of joint works with Alberto Bressan, Helge Kristian Jenssen, Robin Young, Ping Zhang, Qingtian Zhang and Yuxi Zheng.

## Jan 30, David Herzog, Duke University

Abstract: With broad interests in mind, we discuss certain, explosive ODEs in the plane that become stable under the addition of noise. In each equation, the process by which stabilization occurs is intuitively clear: Noise diverts the solution away from any instabilities in the underlying ODE. However, in many cases, proving rigorously this phenomenon occurs has thus far been difficult and the current methods used to do so are rather ad hoc. Here we present a general, novel approach to showing stabilization by noise and apply it to these examples. We will see that the methods used streamline existing arguments as well as produce optimal results, in the sense that they allow us to understand well the asymptotic behavior of the equilibrium measure at infinity.

## Jan 31, Yizao Wang, U. Cincinnati

Abstract:Maximal set-indexed sums provide a general framework for different types of problems. This includes the longest sequence of heads in an coin tossing experiment, the change-point detection in sequential analysis, scan statistics, maximal displacement of a branching random walk, and last passage percolation, among others. The limit laws of these models have attracted many attentions in recent years.

In this talk, we give a brief review over these problems and their recent developments. Then we focus on how the tail behavior of each single random variable affects the limit laws, and the drastically different asymptotic behavior for light-tailed and heavy-tailed models. Accordingly, we will review Poisson clumping heuristic (Pickands’ method) and weak convergence of order statistics, two well developed techniques in investigating extremes.

Joint work with Zakhar Kabluchko.

## Feb 21, Olav Kallenberg, Auburn University

Abstract: L'evy processes are defined as processes with stationary, independent increments. In contrast to the special case of Brownian motion they may have jump discontinuities, which make the definition of stochastic integrals more subtle. Even more general (and more subtle) is the case of exchangeable processes, which require stochastic integration with respect to general semi-martingales. In this talk I will discuss some invariance properties of such integrals obtained at various points throughout my career.

## Jan 10, Kevin McGoff, Duke University

Abstract: Dynamical systems arise frequently as models of physical and biological systems that evolve over time. Beginning with motivation from time-series experiments of gene expression data, I will discuss several probabilistic and statistical properties of dynamical systems, as well as the role these properties play in statistical inference. Lastly, I will discuss recent work in this area related to gene regulatory networks.

## Dec 12, Katya Krupchyk, University of Helsinki

Abstract: More than 25 years ago, Kenig, Ruiz, and Sogge established uniform $L^p$ resolvent estimates for the Laplacian in the Euclidean space. Taking their remarkable estimate as a starting point, we shall describe more recent developments concerned with the problem of controlling the resolvent of elliptic self-adjoint operators in $L^p$ spaces in the context of a compact Riemannian manifold. Here some new interesting difficulties arise, related to the distribution of eigenvalues of such operators. Applications to inverse boundary problems and to the absolute continuity of spectra for periodic Schr\"odinger operators will be presented as well.

## Dec 9, Hantaek Bae, UC Davis

Abstract: It is important to investigate regularity and long-time behavior of the Navier-Stokes equations in connection with turbulence. Turbulence is closely related to length scales, which can be measured in term of the radius of analyticity of solutions. A decisive role is played by Gevrey regularity that provides the radius of analyticity. This approach has several applications in the study of long term dynamics. We here show decay rate of weak solutions and Log-Lipschitz regularity of mild solutions of the 3D Navier-Stokes equations.

## Nov 15, Veronica Martinez de la Vega, Instituto de Matematicas, Universidad Nacional Autonoma de Mexico

Abstract:

A continuum is a compact connected metric space. Among them, dendroids are arcwiseconnected, hereditarilly unicoherent continua. In fact, dendroids can be embedded in $\mathbb{R}^{3}$. For over seventy years, dendroids have been the object of study of many specilists. Though easily defined, there are many interesting results concerning them, and more over, many open problems still unsolved. In this talk I will give some examples, results and problems concerning them.

## Jan 24, Luke Oeding, Auburn

Abstract: Suppose you were at the symphony and I challenged you to put on a blindfold and identify how many instruments were being played just by the sound. Suppose I further asked you to reproduce what single instruments were playing even while the entire symphony was playing different things. How would you do it? If you represent sound with a large multidimensional data set, the mathematical version of theses problems centers on something called tensor decomposition.

Using this musical scenario as motivation, I'll discuss some of the algebraic and geometric aspects of tensor rank, tensor decomposition, and what mathematicians can do to help other scientists in understanding large data sets.

## Feb 28, Sabine Jansen, Bochum

We consider a system of point particles in a finite box in R2 that interact via a finite-range attractive pair potential, and move according to a Markov process that has the grand canonical Gibbs measure as a reversible measure. The chemical potential is such that the system favors a packed box, but has a nucleation barrier to overcome in order to go from an empty box to a packed box. We are interested in the nucleation time in the limit as the temperature tends to zero. We use the potential-theoretic approach to metastability. The results extend earlier work for lattice systems; the main difficulty lies in understanding the energy landscape of the continuum particle system, a problem of intrinsic interest in analysis. The talk reports on joint work in progress with Frank den Hollander (Leiden University, Netherlands).

## Oct 25, Martin Schmoll, Clemson

Recently rigorous results on the periodic wind-tree model have been shown using sophisticated methods provided by Teichmueller Theory. Most of those methods apply to other infinite surfaces with polygonal structures. In this presentation we use some of those methods to study direction foliations on Panov planes and closely related dynamical systems. A Panov plane is the universal cover of a torus. The torus comes equipped with a quadratic differential defining the direction foliations. We describe the relation of the periodic wind-tree model to certain Panov planes and state the central result on their dynamics. Further we mention an application of Panov dynamics to retro-reflector systems (with Krzysztof Fraczek).

Most of the presentation is ongoing research with my graduate student Chris Johnson.

## Nov 8, David Nualart, Kansas

Abstract: The fractional Brownian motion is a centered self-similar

Gaussian process with stationary increments, which depends on a

parameter H in (0,1) called the Hurst index. We will first describe

some basic properties of the fractional Brownian motion such as

long-range dependence and finite p-variation. The applications of the

fractional Brownian to model data coming from engineering, finance and

other areas, require the construction of a suitable stochastic

calculus, similar to the classical Ito calculus. In this talk we

review some recent results on the stochastic calculus with respect to

the fractional Brownian motion with emphasis on the construction of

stochastic integrals using different types of Riemann sums

approximations. We will present central limit results for critical

values of the Hurst parameter where the approximation diverges, and we

will discuss numerical approximation schemes for stochastic

differential equations driven by a fractional Brownian motion with

Hurst parameter H>1/2.

## Sept 6, Mohsen Bayati, Stanford

Reconstructing a high-dimensional sparse vector from a small number of observations is a well-studied problem in many scientific and engineering disciplines, and a number of tools have been designed to address this problem. Recently, Donoho, Maleki and Montanari introduced approximate message passing (AMP) as an extremely effective algorithm for solving this problem. They also showed (through extensive numerical experiments) that dynamics of AMP is accurately tracked by a simple one-dimensional iteration termed "state evolution". We provide a rigorous foundation to state evolution and prove that it holds asymptotically for random matrices. Using our analysis, we obtain rigorous expressions for the risk of LASSO.

If time permits, I will discuss applications of these methods in clinical decision making.

This is joint work with Jose Bento, Marc Lelarge, and Andrea Montanari