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 lowlying 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
sinekernel 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 nonlinear 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 minicourse 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 threedimensional 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 minicourse 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 twodimensional Schrödinger operators with random δmagnetic 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 46 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 onedimensional 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
•
ShortLong 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 FirstPassage Percolation Across Thin Cylinders

We consider firstpassage percolation on the graph Z x {h_n,h_n+1,...,h_n}^{d1}
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 firstpassage time between two
vertices is defined as the minimum passage time over all paths joining the two vertices. We
will show that the firstpassage time T_{n} 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 T_{n}
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 RiemannHilbert problems

Using the Fourier transform, solutions of a linear PDE with constant coefficients can be
written as oscillatory integrals whose longtime asymptotics can be studied using the
stationary phase method. A RiemannHilbert problem is a factorization problem and it can be
used to invert the onedimensional 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 RiemannHilbert problems, which can be used to obtain
longtime 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 limitperiodic.

Mark Herman
•
Convergence Issues of MöllerPlesset Perturbation Theory in Heliumlike Atoms

MöllerPlesset 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
MöllerPlesset 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
onedimensional model with delta functions in place of Coulomb potentials and the realistic
threedimensional 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 MeanField Limit

It is well known that the meanfield 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 onebody Hamiltonians and interaction potentials. In particular, we
control the quantum fluctuations in the meanfield 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 n^{ρ})
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 l^{p} condition and a generalized bounded variation condition. Examples of
sequences satisfying this condition are linear combinations of cos(nφ+α) n^{γ}
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 HalfLine 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
σfinite measure ν. It is examined how various degrees of continuity of ν with respect
to αdimensional 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 manybody wave function, as
the time approaches the collapse time.

Stephen Ng
•
Ferromagnetic Ordering of Energy Levels

We show that the Heisenberg model on onedimensional, 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 x_{k} denote eigenvalue number k. Under the condition
that both k and n−k tend to infinity with n, we show that x_{k} 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 nonGaussian entries that have an exponentially decaying
distribution and whose first four moments match the Gaussian moments.

Luc ReyBellet
•
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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
qorthogonal polynomials and point out their relevance in the study of some topological
gauge theories.

Aaron Welters
•
On Constructibility Results for a Class of NonSelfadjoint 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 nonselfadjoint analytic perturbations of matrices
with degenerate eigenvalues. Based on some mathphysics 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 finding explicit recursive formulas can be solved. Using these recursive
formulas, I will list the first 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 X_{1}, X_{2} and X_{3} 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 onedimensional Isingtype 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 RiemannHilbert Problems in Supercavitation

In this talk I will consider a problem for a supercavitating wedge under a free surface
with the Tulin singlespiralvortex cavity closure condition. The mechanical problem will be
reduced to two RiemannHilbert problems on an elliptic Riemann surface. The solution of these
problems will be discussed. The numerical results will be presented.
