**
Eric Carlen (Rutgers University):
***Functional Inequalities and Gradient Flow for Quantum Markov Semigroups*
In recent years, building on work of Felix Otto, much progress has been made in the study of
a wide class of evolution equations for probability densities by viewing them as gradient flow for certain entropy functions with respect to mass-transportation metrics. The simplest example is the classical Fokker-Planck equation, which was shown by
Jordan, Kinderleher and Otto to be gradient flow in the 2-Wasserstein metric for the relative entropy with respect to the steady-state Gaussian density. The Fokker-Planck equation has several natural quantum analogs, in particular one for fermions. This has the form of a Lindblad evolution equation for a time-dependent density matrix. There is a natural differential structure that allows this equation to be written as a "non-commutative partial differential equation", and also, as was shown by myself and Jan Mass, to define a natural analog of the 2-Wasserstein distance as a Riemannian distance on the manifold of density matrices such that the equation is, as in the classical case, gradient flow in this metric for the relative entropy with respect to the ground state. Recent joint work with Jan Maas has extended this to a wide class of quantum evolutions equations, linear and non-linear. As in the classical case, a wide range of functional inequalities governing the evolution can be seen as consequences of convexity with respect to the underlying transport metric. These lectures will provide an introduction form the beginning to quantum evolution equations and gradient flow. The close parallels with the classical theory will be emphasized, and a number of open problems will be pointed out.

**
Fritz Gesztesy (Baylor University):
***Eigenvalue counting function bounds for the Krein-von Neumann extension associated with uniformly elliptic PDEs.
*
We derive a bound for the eigenvalue counting function for Krein-von Neumann extensions corresponding to a class of uniformly elliptic second order pde operators (and their positive integer powers) on arbitrary open, bounded, n-dimensional subsets $\Omega \subset \mathbb{R}^n$. No assumptions on the boundary of $\Omega$ are made.

Our technique relies on variational considerations exploiting the fundamental link between the Krein-von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of the corresponding differential operator suitably extended to all of
$\mathbb{R}^n$.

We also consider the analogous bound for the eigenvalue counting function for the corresponding Friedrichs extension.

This is based on joint work with M. Ashbaugh, A. Laptev, M. Mitrea, and S. Sukhtaiev.

**Michael Goldstein (Toronto University):
***On localization and spectrum of multi-frequency quasi-periodic operators. Report on recent progress*
This talk will report on results on multi-frequencies quasi-periodic operators developed in recent joint works and ongoing projects with W.Schlag and M.Voda.

**
Anton Gorodetski (University of California Irvine):
***Random matrix products with parameter*
Random products of matrices appear naturally in many different settings, in particular in smooth dynamical systems, probability theory, spectral theory, mathematical physics. The crucial result is Furstenberg's Theorem on positivity of Lyapunov exponents. It claims that generically the exponential rate of growth (Lyapunov exponent) of product of random matrices is well defined and positive. In the talk we will discuss the random products of 2x2 matrices that depend on a parameter. This is motivated, in particular, by the study of discrete Schrodinger operators with random potentials. In that case the Schrodinger cocycle is given by the random products of transfer matrices, and energy serves as a natural parameter. From spectral point of view it is natural to fix the potential first, and then vary the energy. As a more general setting, one can consider random products of matrices depending (monotonically) on a parameter, and study existence and properties of Lyapunov exponent for a typical fixed sequence when the parameter varies. It turns out, for example, that in the non-uniformly hyperbolic regime almost surely upper Lyapunov exponent is positive (and coincides with the one prescribed by Furstenberg Theorem) for all parameters, but lower Lyapunov exponent vanishes for a topologically generic parameter. These results explain the difficulties one encounters in the classical proofs of Anderson localization for random Schrodinger operators. This is a joint project with V. Kleptsyn.

**
Philip Gressman (University of Pennsylvania):
***Geometric Averages in Harmonic Analysis*
I will present recent results relating to the problem of quantifying L^p-improving properties of convolutions with singular measures. This theory is much more complete for measures supported on curves and hypersurfaces than it is for submanifolds of intermediate dimension. This relative lack of positive results is due in part to the problem that the Phong-Stein rotational curvature condition (which governs nondegeneracy in many important cases) is frequently impossible to satisfy for surprisingly deep algebraic reasons. I will focus primarily on the case of 2-surfaces in R^5, which does not fit nicely into previously-existing combinatorial strategies, and will present a new approach with the potential to apply in a broad range of new situations.

**
Michael Loss (Georgia Tech):
***Modeling thermostats using Master equations***
**
In this talk we discuss results for a model of randomly colliding particles
interacting with a thermal bath, i.e., a thermostat. Collisions between particles are modeled via
the Kac master equation while the thermostat is seen as an infinite gas at
thermal equilibrium with inverse temperature $\beta$. The evolution propagates
chaos and the one particle marginal, in the limit of large systems, satisfies an effective Boltzmann-type equation. The system admits the
canonical distribution at inverse temperature $\beta$ as the unique equilibrium
state. It turns out that any initial distribution approaches the equilibrium
distribution exponentially fast, both, in a proper
function space as well as in relative entropy. Recent results concerning the approximation
of thermostats by a large but finite heat reservoir will also be discussed. It turns out that in suitable
norms the approximation can be shown to be uniformly in time, i.e., the error depends only
on the size of the finite heat reservoir.
This is joint work with Federico Bonetto, Hagop Tossounian and Ranjini Vaidyanathan.

**
Lai Sang Young (Courant Institute):
***Nonequilibrium steady states of certain particle systems*
This talk is about a class of particle systems in which energy exchange
among particles is mediated by the environment, symbolized by arrays of
rotating disks in mechanical models and energy tanks in stochastic models.
A number of years ago, J-P Eckmann and I carried out a semi-rigorous study
of models of this type, deriving formulas for energy and particle density
profiles for chains coupled to unequal heat baths. In this earlier work, we
took for granted analytical issues such as existence and uniqueness of NESS,
local thermal equilibria, etc. More recently I revisited some of these issues,
and obtained partial resolution for some of them. In this talk I will review
earlier results and present new ones.

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