## Colloquium 2013-2014

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

Colloquium CH458, 2:30pm. Ergodicity of the spin-boson model for arbitrary coupling strengthI 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

Colloquium 458 Campbell Hall, 2:30pm. The dimer model on the hexagonal lattice.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

Colloquium 458 Campbell Hall, 2:30pm. On the Local Behavior of Non-Negative Solutions to a Logarithmically Singular Equation.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

Colloquium, 458 Campbell Hall, 9:30am. Slowly decaying perturbations of Schrodinger and Jacobi operators.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

Colloquium 458 Campbell Hall, 10:10am. Singular behavior in the solutions of nonlinear hyperbolic PDEsAbstract: 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

Colloquium 458 Campbell Hall, 9:30-10:30am. Noise-Induced Stabilization of Planar Flows.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

Colloquium: 458 Campbell Hall, 2:30-3:30pm.Limit laws of maximal set-indexed sums.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

Colloquium, 458 Campbell Hall, 2:30-3:30pm. Some invariance properties involving stochastic integrals with respect to L'evy and related processesAbstract: 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

Colloquium, 458 Campbell Hall, 11:15AM. Stochastic properties of dynamical systems.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

Colloquium, 458 Campbell Hall, 9:30am. Resolvent estimates for elliptic operators and their applications.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

Colloquium, 458 Campbell Hall, 9-10AM. Gevrey regularity of the Navier-Stokes equations and applications.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

Colloquium, 458 Campbell Hall, 2:30-3:30, Dendroids: Examples, Results and Problems.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

Colloquium: 458 Campbell Hall, 2:30-3:30pm. Tensor decomposition and applications.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

Colloquium, 458 Campbell Hall, 2:30-3:30. Metastability for continuum interacting particles.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

Colloquium, 458 Campbell Hall, 2:30-3:30. Dynamics on Panov planes and applicationsRecently 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.

## April 4, Jason Swanson, Central Florida

Colloquium, 443 Campbell Hall, 2:30-3:30pm. (Sping Tea for graduates following.)Title: Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion.

Abstract: The signed cubic variation of fractional Brownian motion (fBm) is obtained by considering the sum of the cubes of the increments of fBm over uniformly spaced time intervals whose lengths, t, tend to zero. It is well-known that when the Hurst parameter of fBm is set to H = 1/6, this sum converges in distribution to a Brownian motion that is independent of the original fBm. In this talk, I will discuss recent joint work with Chris Burdzy and David Nualart in which we study the asymptotic correlation between two distinct sums of this type, where the diff erence between the two sums is in the value of t.

## Nov 8, David Nualart, Kansas

Colloquium, 458 Campbell Hall, 2:30-3:30. Stochastic calculus with respect to the fractional Brownian motion.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.

## Dec 6, Jason Starr, Stony Brook

Colloquium, 458 Campbell Hall, 2:30-3:30PM. Solving polynomials, finding fixed points, splitting fibrationsAbstract: There are strong analogies between the problem of finding fixed points for group actions on a manifold, finding sections of a fibration of topological spaces, and finding solutions of systems of polynomial equations. In particular, every fibration over a curve whose fibers are path connected admits a section, as does every fibration over a surface whose fibers are simply connected. There are algebro-geometric analogues of path connected and simply connected: "rationally connected" and "rationally simply connected" complex projective manifolds. Work of Graber, Harris and myself and of de Jong, He and myself establishes analogues of the fibration result in algebraic geometry. Chenyang Xu and I combined this with beautiful work of Helene Esnault to explain several classical results in number theory (over global function fields): the Tsen-Lang theorem, a theorem of Brauer-Hasse-Noether, and the split case of a theorem of Harder on Serre's "Conjecture II".

## Mar 14, Jonathan Pakianathan, Rochester

Colloquium, 458 Campbell Hall, 2:30pm. An Elementary Construction of Tesselated Riemann Surfaces.Tesselated polyhedra have been studied from the beginnings of mathematics. Regular spherical polyhedra like those given by the Platonic solids have long been connected to aspects of group theory through the study of reflection groups and through Fuchsian group actions on the upper half plane. However these connections require a decent amount of machinery. In this talk, I will outline an elementary and natural construction studied by Mark Herman, Ergun Yalcin and myself which constructs a singular surface naturally from any finite nonabelian group. This surface desingularized yields a collection of tessellated Riemann surfaces with regular or near-regular tessellation from which subquotients of the group act as the object's automorphism group. The key aspect of the construction is that the group algorithmically constructs these surfaces in a direct manner together with their tesselations and action. No inspiration is needed to find them and the constructions yields many famous classical examples together with potentially new ones. A single group produces a large collection of tessellated surfaces encoding the algebraic structure of the group - for example, the symmetric group on 6 letters produces 4477 tessellated surfaces of 27 distinct genus. Many examples will be discussed during the talk.

## Nov 22, Wei-Kuo Chen, Chicago

Colloquium, 458 Campbell Hall, 2:30-3:30, On Gaussian inequalities for product of functions.Abstract: Gaussian inequalities have played important roles in various scientific areas. In this talk, we will present simple algebraic criteria that yield sharp Holder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. As an application, we will explain how our results yield several famous inequalities in functional geometry, such as, the Brascamp-Lieb inequality, the sharp Young inequality, etc. This part of the talk is based on the recent joint work with N. Dafnis and G. Paouris. Along this direction, we will discuss a conjecture on the convexity of the Parisi functional arising from the study of the Sherrington-Kirkpatrick model in spin glass.

## Oct 18, Thierry Colin, Bordeaux

Mathematical models for tumor growth: construction, validation and clinical applications. Campbell Hall 458. 2:30-3:30Abstract:

In the last few years there have been dramatic increases in the range and quality of information available from non-invasive medical imaging methods, so that several potentially valuable imaging measurements are now available to quantitatively measure tumor growth, assess tumor status as well as anatomical or functional details. Using different methods such as the CT scan, magnetic resonance imaging (MRI), or positron emission tomography (PET), it is now possible to evaluate and define tumor status at different levels: physiological, molecular and cellular.

These multimodal data help the decision process of oncologists in the definition of therapeutic protocols. At present, this decision process is mainly based on previously acquired statistical evidence and on the practitioner experience. The quality of the response to a treatment is decided according to the OMS criteria by estimating the length of the two main axis of the tumor in the largest cut. There are two blocking difficulties in this approach that we want to attack: i) previous statistical information is not patient specific; ii) there exist no quantitative mean of summarizing and using as predicting tools the multimodal patient-specific data presently available thanks to CT scans, MRI, PET scans and molecular biology data. The aim of this talk is to show how we can provide a simulation framework based on quantitative patient-specific data by using nonlinear model based on PDE. I will present how one can build such model in order to describe the may features of tumor growth and how one can expect to obtain a suitable parametrization of tumor growth by solving inverse problem. I will present some mathematical results on the models.

The applications will concern lung and lever metastasis, meningiomas and brain tumor.

## Oct 11, Thomas Schultz, Bonn

Colloquium: Symmetric Tensor Approximation and the Human Brain Connectome. 458 Campbell Hall. 2:30-3:30pm.

Mapping the human brain connectome is a major scientific effort that requires combining cutting-edge technology - in particular, specific variants of Magnetic Resonance Imaging (MRI) - with suitable mathematical theory for data analysis. This talk highlights how symmetric tensors and their low-rank approximation or, equivalently, homogeneous polynomials and their approximation with a sum of powers of a few linear forms, provide a state-of-the-art solution to the crossing fiber problem, one of the fundamental challenges in mapping the brain connectome from MRI data. I will cover theoretical and algorithmic aspects of this framework, and touch upon how blind deconvolution based on sparsity assumptions can further refine it.

## Oct 7, Carl Mueller, Rochester

Colloquium: Hitting questions for Gaussian processes and stochastic partial differential equations. 458 Campbell. 2:30-3:30pm.Abstract: I will begin by giving an overview of why hitting questions are among the most intensely studied topics in stochastic processes. For example, if we manage a portfolio and the losses and gains are random, we may be interested in whether the portfolio value hits 0 (and we go broke). Or, considering point processes, we might be interested in the amount of time it would take to see a rare event such as a US debt default. There is one case in which the answer is completely known, at least in principle: for Markov processes. However, many random fields and other processes are not Markov. A lot is also known about hitting questions for Gaussian processes, although not as much as for Markov processes. The new results in this talk involve work in progress, which extends some of the results about hitting questions for Gaussian processes to nonlinear stochastic partial differential equations. This work ties in with the following longstanding problem: to what extent do the special properties of Gaussian processes carry over to other random fields?

## Oct 4, William T. Mahavier, Lamar University

Colloquium: IBL Calculus (Two very different approaches). 458 Campbell Hall, 2:30-3:30pm.

Abstract: We discuss two approaches to teaching an IBL Calculus course. The first is a gentle step away from the lecture method and the second is a more intense version of IBL. We will address the resources available to those interested in implementing IBL instruction in their classrooms as well as literature that addresses efficacy of the method.

## Sept 27, Lek-Heng Lim, Chicago

Colloquium: Tensors and Hypermatrices. 458 Campbell Hall 2:30-3:30pm.

Abstract: This talk is intended for those who, like the speaker, have at some point wondered whether there is a theory of three- or higher- dimensional matrices that parallels matrix theory. We will discuss how notions like rank, norm, determinant, eigen and singular values may be generalized to hypermatrices. We will see that, far from being artificial constructs, these notions have appeared naturally in a wide range of applications: chemistry (fluorescence spectroscopy, density matrix renormalization group), computer science (matrix multiplication complexity, quantum computing), optimization (self-concordance, higher-order optimality conditions), statistics (higher-order moments and cumulants, minimum rank matrix completion), physics (quark states, Yang-Baxter equations), and signal processing (antenna array processing, CDMA radio communication).

## Sept 20, Antonio Auffinger, Chicago

Colloquium Title: Geodesics in First Passage Percolation. 458 Campbell Hall 2:30-3:30First-passage percolation is a model of a random metric on a infinite network. It deals with a collection of points which can be reached within a given time from a fixed starting point, when the network of roads is given, but the passage times of the road are random. It was introduced back in 60's but most of its fundamental questions are still open. In this talk, we will overview some recent advances in this model focusing on the existence, fluctuation and geometry of its geodesics. Based on joint works with M. Damron and J. Hanson.

## Sept 6, Mohsen Bayati, Stanford

Colloquium: Analysis of Approximate Message Passing and the Risk of LASSO. 458 Campbell Hall 2:30-3:30pm.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