Isoperimetric Inequalities for Eigenvalues of the Laplacian
Rafael Benguria

Isoperimetric inequalities have a long history in mathematics dating back to the Greeks and Dido's problem, i.e., the classical isoperimetric inequality in Euclidean geometry. With the introduction of the Calculus of Variations in the seventeenth century, isoperimetric inequalities found their way into mechanics and physics. In 1856, Adhémar Jean Claude Barré de Saint Venant introduced the problem of determining the shape of the column of a given section having the maximal torsional rigidity, a problem that has important practical implications. Other physical quantities have been the sub ject of interesting isoperimetric inequalities as well, among them the electrostatic capacity. The theme of these lectures, isoperimetric inequalities for eigenvalues of the Laplacian, has its roots in the work of Lord Rayleigh on the theory of sound. It was determined in the nineteenth century that the basic equation that describes the small vibrations of an elastic medium is the wave equation. The normal modes and proper frequencies that characterize the vibrations of a fixed, homogeneous membrane correspond to particular solutions of the wave equation. They are determined by the solution of the eigenvalue problem for the Dirichlet Laplacian on a bounded domain in R^2. The purpose of these lectures is to give an overview of some isoperimetric inequalities for eigenvalues of the Laplacian. Since the literature on the sub ject is extensive, the lectures will be restricted primarily to the consideration of isoperimetric results for low-lying eigenvalues with special attention to those connected to eigenvalue ratios.
 

Universality of Wigner Random Matrices
Laszlo Erdös

Random matrices arise in several areas of mathematics, physics and statistics. Most prominently, since the fundamental observation of E. Wigner, they are used to model quantum systems whose precise Hamiltonian is unknown. Their spectral statistics show fascinating universality properties, most notably the Wigner semicircle law for the density of states and the determinental sine-kernel law for the local correlation of eigenvalues. These laws are independent of many details of the underlying probability distribution.

The Wigner random matrix ensemble consists of matrices in a certain symmetry class (hermitian or symmetric) whose entries are  identically distributed random variables with maximal independence allowed by the symmetry. If the distribution is Gaussian (GUE or GOE ensembles), then several explicit formulae are available to compute spectral statistics. Such formulae are not present for a general distribution. In the recent years several  new analytical tools have been developed to tackle the general case and thus proving universality for a broad class of Wigner matrices.

The main goal of the lectures is to show that hermitian or symmetric Wigner matrices with a general single site distribution follow the same local spectral statistics as GUE or GOE. The core of the argument is a hydrodynamical approach that is concise and  will be presented in full details. The random matrix will be imbedded into a stochastic flow and the corresponding eigenvalue process will be studied. It turns out that universality of the Gaussian ensembles can be considered as relaxation to equilibrium if this flow of eigenvalues. To apply this theory, several preparatory results will be presented, most importantly I will discuss the proof of the local version of the semicircle law.

Random matrices can also be viewed as a mean field version of the random Schrödinger operators. Apart from the spectral statistics, analogues of localization/delocalization  and Wegner type estimates will be discussed.

The lectures assume no previous knowledge about random matrices. Some familiarity with elementary probability theory is helpful.
 

Kinetic Theory and Kac’s Master Equation
Michael Loss

Many natural phenomena are described in terms of kinetic ideas, i.e., in terms of 'particles' that interact via 'collisions'. These 'particles' may be objects which coagulate, whose dynamics is described by Smoluchowski's equation, or birds that flock and interact with each other by avoiding collisions. The most famous example of kinetic theory is, of course, Boltzmann's equation. Mathematically, such systems pose considerable challenges. The derivation of Boltzmann's equation from the fundamental laws of mechanics is still not well understood. Already the spatially homogeneous Boltzmann equation, being a non-linear integral equation, poses difficult problems in analysis. In the mid fifties, Mark Kac proposed a Master Equation describing collisions of particles. This is a linear stochastic evolution for the probability distribution of the velocities of N particles. By introducing a notion, now called propagation of chaos, he was able to derive the spatially homogeneous Boltzmann equation in the limit as the particle number N tends to infinity. The aim of these lectures is to demonstrate that such equations are quite tractable. In particular, the gap, which gives a rate for the approach to equilibrium, can be calculated exactly. Further, one has a fairly good understanding of what the chaotic states look like. In the mini-course the connection between the Master Equation and the spatially homogeneous Boltzmann Equation will be explained as well as the calculations of the gap for one- and three-dimensional collision models. The methods are quite simple and use Gegenbauer and Jacobi polynomials. As it turns out, these methods have something to say about orthogonal polynomials, in particular they give relatively painless proofs of some deep results of Gegenbauer, Bochner and Gasper on the Markov Sequence Problem.
 

Localization in Disordered Media
Günter Stolz

Disorder has profound effects on electronic transport. In a disordered material, such as an alloy, all electron wave functions in a certain energy range can become dynamically localized, resulting in zero conductivity. A half century ago P.W. Anderson introduced a model which allowed to explain this, first physically and later with mathematical rigor. The model and phenomenon are now named after him: Anderson model and Anderson localization. While other models have been used, the Anderson model is still the prototypical and most studied example of a random Schrödinger operator, in this case a discrete Schrödinger operator. Various ways have been found in which Anderson localization can be expressed. Dynamical localization can be established directly in terms of bounds on spatial spreading for solutions of the time dependent Schrödinger equation. Closely related to this is the phenomenon of dense point spectrum. Two methods have been found which provide mathematical proofs of these localization properties in arbitrary space dimension. In 1983 Fröhlich and Spencer introduced the method of multiscale analysis, which lead to the first proofs of multidimensional Anderson localization. One decade later Aizenman and Molchanov found the fractional moments method which has provided remarkably straightforward proofs of localization, including a quite strong form of dynamical localization. This mini-course will give an introduction to the mathematical theory of Anderson localization based on the fractional moments method. A complete proof of spectral and dynamical localization for the Anderson model at large disorder will be presented. It is also planned to describe extensions of the method which allow to show localization results for continuum analogs of the Anderson model and, quite recently, for systems of interacting electrons in an external random potential.
 

Regular talks

Alain Joye  •  Dynamical Localization for Unitary Anderson Models
To begin with, the construction of unitary Anderson models will be described and motivated by physical considerations. Then, the dynamical localization properties of these models will be addressed in the following three regimes: arbitrary disorder in dimension one, strong disorder in higher dimension and arbitrary disorder in higher dimension at band edges. We will pay a particular attention to this last regime. If time permits further applications of these ideas to random quantum walks will be discussed. Based on joint works with Eman Hamza, Günter Stolz and Marco Merkli.

Takuya Mine  •  Lifshitz Tail for Schrödinger Operators with Random δ Magnetic Fields
The Lifshitz tail (the exponential decay of the integrated density of states near the edge of the spectrum) is one of the fundamental estimates in the theory of Anderson localization, and is mathematically proved for many types of Schrödinger operators with random scalar potentials. However, there are not so many results for the Schrödinger operators with random magnetic fields. In this lecture, we shall prove the Lifshitz tail for two-dimensional Schrödinger operators with random δ-magnetic fields.

Simone Warzel  •  Localization on trees at weak disorder?

 

Short talks

The main goal of the short talks is to have a panorama of the current interests of participants. The duration will be 4-6 hours, divided by the number of volunteers -- probably around 10 minutes. The sessions of short talks should be fun to attend, and these may help stimulate discussions. The organizers are very grateful to all people volunteering for short talks.

Antonio Auffinger  •  Complexity of Random Morse Functions and Random Matrix Theory
We relate the computation of the complexity of general Gaussian functions on the Sphere in large dimensions with random matrix theory. Using this link we compute their complexity, i.e their number of critical points, of given index on any level sets. We relate this question to the description of the the low energy states of spherical spin glasses. This joint work with G. Ben Arous and J. Černý.

Sven Bachmann  •  The Anderson Model on a Strip and Random Matrix Theory
Starting from the Anderson model on a quasi one-dimensional strip, I shall show that a twofold scaling limit allows to recover the fundaments of the random matrix theory that accounts for the universal metallic properties of disordered wires, the DMPK theory.

Shirshendu Chatterjee  •  Short-Long First Passage Percolation on Two Dimensional Torus

Ivan Corwin  •  Fluctuations of Last Passage Percolation
I will review recent work on the fluctuation theory of directed last passage percolation with boundary conditions. This model relates to the corner growth model with external sources.

Partha Dey  •  Central Limit Theorem for First-Passage Percolation Across Thin Cylinders
We consider first-passage percolation on the graph Z x {-h_n,-h_n+1,...,h_n}d-1 where each edge has an i.i.d. nonnegative weight. The passage time for a path is defined as the sum of weights of all the edges in that path and the first-passage time between two vertices is defined as the minimum passage time over all paths joining the two vertices. We will show that the first-passage time Tn between the origin and the vertex (n,0,...,0) satisfies a Gaussian CLT as long as h_n=o(n^α) with α < 1/(d+1). The proof will be based on moment estimates, a decomposition of Tn as a sum of independent random variables and a renormalization type argument. This is a joint work with Sourav Chatterjee.

Yen Do  •  A stationary phase method for oscillatory Riemann-Hilbert problems
Using the Fourier transform, solutions of a linear PDE with constant coefficients can be written as oscillatory integrals whose long-time asymptotics can be studied using the stationary phase method. A Riemann-Hilbert problem is a factorization problem and it can be used to invert the one-dimensional scattering transform, which is a nonlinear Fourier transform for many nonlinear PDEs. I will describe a nonlinear analogue of the classical stationary phase method for oscillatory Riemann-Hilbert problems, which can be used to obtain long-time asymptotics for solutions of such nonlinear PDEs.

Zheng Gan  •  Optimality of Log Hölder Continuous of the Integrated Density of States
We construct examples of Schrödinger operators, that log Hölder continuity of the integrated density of states cannot be improved. Our examples are limit-periodic.

Mark Herman  •  Convergence Issues of Möller-Plesset Perturbation Theory in Helium-like Atoms
Möller-Plesset perturbation theory is a widely used method in computational chemistry to approximate the ground state energy of a molecular system.  Most often, only the first few terms of an infinite series are computed, but little is known as to whether the series converges, and if this approximation is meaningful. We study convergence or divergence of the Möller-Plesset series for systems with two electrons and a single nucleus of charge Z>0. This question is essentially to determine if the radius of convergence of a power series in the complex perturbation parameter λ is greater than 1. We examine a simple one-dimensional model with delta functions in place of Coulomb potentials and the realistic three-dimensional model. For each model, we show rigorously that if the nuclear charge Z is sufficiently large, there are no singularities for real values of λ between -1 and 1. Using a finite difference scheme, we present numerical results for the delta function model.

Antti Knowles  •  Controlling Quantum Fluctuations in the Mean-Field Limit
It is well known that the mean-field time evolution of a quantum Bose gas is governed by the nonlinear Hartree equation. I will report on recent joint work with Peter Pickl, in which we derive explicit bounds on the magnitude of the quantum fluctuations. Our results cover a large class of one-body Hamiltonians and interaction potentials. In particular, we control the quantum fluctuations in the mean-field approximation of a boson star, and provide a microscopic derivation of the nonrelativistic Hartree equation with a critical nonlinearity. We also show that in a scattering regime the quantum fluctuations are bounded uniformly in time.

Rostyslav Kozhan  •  Meromorphic Continuations of Finite Gap Herglotz Functions
Let μ be a finite measure on the real line whose essential support E is a finite number of intervals. Then the associated Herglotz function m is meromorphic on \mathbb{C}\E. \mathbb{C}\E can be viewed as one of the two sheets of a natural Riemann surface. I find a necessary and sufficient condition for the function m to have a meromorphic extension to some region on the second sheet of this surface. The domain of meromorphicity can be described explicitly in terms of the distance of the associated Jacobi matrix from the isospectral torus (of periodic Jacobi matrices) corresponding to the set E.

Helge Krüger  •  Schrödinger operators with potential V(n) = λ f(α nρ)

Milivoje Lukic  •  Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation
We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an lp condition and a generalized bounded variation condition. Examples of sequences satisfying this condition are linear combinations of cos(nφ+α) n with γ>0. We prove that such measures have no singular continuous part, that the support of their a.c. part is the same as for the corresponding free case, and that there is at most a finite number of embedded pure points.

Anna Maltsev  •  Universality Limits of a Reproducing Kernel for a Half-Line Schrodinger Operator and Clock Behavior of Eigenvalues

Chris Marx  •  Continuity of Spectral Averaging
We consider averages κ of spectral measures of rank one perturbations with respect to a σ-finite measure ν. It is examined how various degrees of continuity of ν with respect to α-dimensional Hausdorff measures (0 \leq α \leq 1) are inherited by κ. This extends Kotani's trick where ν is simply the Lebesgue measure.

Alessandro Michelangeli  •  Dynamical Description of Gravitational Collapse
I shall report on a recent joint work with B. Schlein. We consider a semirelativistic system of gravitating particles (boson star). We monitor the occurrence of its gravitational collapse providing an effective control on the loss of regularity of the many-body wave function, as the time approaches the collapse time.

Stephen Ng  •  Ferromagnetic Ordering of Energy Levels
We show that the Heisenberg model on one-dimensional, finite lattices exhibit an ordering of energy levels based on the total spin of the system and give certain conditions under which ordering of energy levels carries over to higher spin generalizations.

Sean O'Rourke  •  Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an n×n matrix from the Gaussian orthogonal ensemble (GOE) and let xk denote eigenvalue number k. Under the condition that both k and n−k tend to infinity with n, we show that xk is normally distributed in the limit.  Using a recent universality result by Tao and Vu, we extend our results to a class of Wigner real symmetric matrices with non-Gaussian entries that have an exponentially decaying distribution and whose first four moments match the Gaussian moments.

Luc Rey-Bellet  •  tba

Arnab Sen  •  Geometric Influences

Shannon Starr  •  Remark on the Central Limit Theorem and Random Analytic Functions
We consider random power series with independent random coefficients with mean 0 and variances appropriate for: the plane, sphere or hyperbolic plane. We note that the CLT may be applied to translates along geodesics (simultaneously dilating in the spherical case).

Miguel Tierz  •  On Random Matrices with Weakly Confining Potentials
We first give an introduction to random matrix theory, emphasizing the orthogonal polynomials approach. Then, we present a family of random matrices characterized by weakly confining potentials. We also show that these models can be solved with q-orthogonal polynomials and point out their relevance in the study of some topological gauge theories.

Aaron Welters  •  On Constructibility Results for a Class of Non-Selfadjoint Analytic Perturbations of Matrices with Degenerate Eigenvalues
In this talk, I consider the problem of finding explicit recursive formulas to compute the perturbed eigenvalues and eigenvectors of non-selfadjoint analytic perturbations of matrices with degenerate eigenvalues. Based on some math-physics problems arising from the study of slow light in photonic crystals, we single out a class of perturbations that satisfy what I call the generic condition. It will be shown that for this class of perturbations, the problem mentioned above of fi nding explicit recursive formulas can be solved. Using these recursive formulas, I will list the fi rst and second order terms for the perturbed eigenvalues and eigenvectors of perturbations belonging to this class.

Peter Windridge  •  Optimal Polling of Three Markov Voters
Suppose that X1, X2 and X3 are discrete Markov processes on a space containing absorbing states "yes" and "no". Eventually, either at least two processes are absorbed at "yes" or two at "no" and so determine a majority decision. We wish to find this decision but are only allowed to reveal the processes one step at a time. Problem: find the majority decision as quickly as possible.

Meng Xu  •  Convergence of Particle Filtering Method for Nonlinear Estimation of Vortex Dynamics
In this talk I will formulate a numerical approximation method for the nonlinear filtering of vortex dynamics subject to noise using particle filtering technique. The convergence of this scheme allowing the observation vector to be unbounded will also be shown.

Mei Yin  •  A Markov Chain Approach to Renormalization Group Transformations
We consider one-dimensional Ising-type Hamiltonian with nearest neighbor interactions and a magnetic field. I will present a Markov chain method for an explicit characterization of the renormalized Hamiltonian after decimation transformation, a commonly used renormalization group (RG) transformation.

Anna Zemlyanova  •  Applications of the Riemann-Hilbert Problems in Supercavitation
In this talk I will consider a problem for a supercavitating wedge under a free surface with the Tulin single-spiral-vortex cavity closure condition. The mechanical problem will be reduced to two Riemann-Hilbert problems on an elliptic Riemann surface. The solution of these problems will be discussed. The numerical results will be presented.