HColl_2012-2013

It is well known that real measures on the circle are characterized by their Herglotz transform, an analytic function in the unit disc. Invariance of the measure under N-multiplication translates into a functional equation for the Herglotz transform. Using elements from the theory of Hardy spaces one gets a somewhat surprising condition for a sequence of complex numbers to be the Fourier coefficients of an N-invariant measure.
Next, starting from any atomless measure on the circle we construct atomless premesures of bounded kappa-variation in the sense of Korenblum which are invariant under s given pairwise prime integers. The relevant function kappa is a generalized entropy function depending on s. The proof uses Korenblum's generalized Nevanlinna theory. Passing to "kappa-singular measures" and extending these to elements in a Grothendieck group of possibly unbounded measures on the circle, one obtains generalized invariant measures which are carried by "kappa-Carleson" sets. The range of this construction depends on interesting questions about cyclicty in growth algebras of analytic functions on the unit disc. If there is time, we also discuss some very formal relations with Witt vectors. For example the Artin-Hasse p-exponential "is" a p-invariant premeasure of bounded
kappa_1 variation.

Patrick Gerard (Universite Paris Sud): Lack of dispersion, integrable systems and inverse spectral problems

Dispersion is a very useful property when studying a large class of evolution partial differential equations, such as nonlinear Schroedinger equations or nonlinear multidimensional wave equations. When this property fails, few methods are available to study the dynamics of the equation. I will present a simple example of such an equation, which unexpectedly ends up at completely integrable dynamics on the Hardy space of the unit disc, connected to inverse spectral problems for Hankel operators. This talk is inspired by works in collaboration with Sandrine Grellier and Herbert Koch.

Michael Harris (Paris): In search of a Langlands transform

Class field theory expresses Galois groups of abelian extensions of a number field F in terms of harmonic analysis on the multiplicative group of locally compact topological ring, the adèle ring, attached to F. Among the most far- eaching predictions of the Langlands program is the existence of a vast generalization of class field theory, in the form of a correspondence between n-dimensional representations of the Galois group of F and automorphic representations, which arise in the harmonic analysis on a homogeneous space for GL(n) of the adèle ring of F. In some cases, the cohomology of Shimura varieties provides a procedure for transforming automorphic representations to Galois representations. I will outline the scope and limitations of this partial realization of the hypothetical Langlands correspondence. The project to use Shimura varieties to construct Galois representations is now nearly complete: I will also describe joint work with Lan, Taylor, and Thorne that constructs a new family of Galois representations that cannot be obtained by any known version of a Langlands transform.

Dmitry Ioffe: Two-dimensional Potts Model

Potts model is a natural generalization of the Ising model. Each site of the lattice receives one of q colours. The colours are sampled from a Gibbs distribution with pair interaction. The interaction is ferromagnetic in the sense that the energy is proportional to the number of frustrated pairs.
It is known that Potts models in two dimensions undergo phase transition. Contrary to the Ising model (q=), however, there are no exact solutions for q>2 Potts models, in particular there are no explicit expressions for quantities like inverse correlation length or surface tension. Hence the need for robust probabilistic tools.
I shall try to explain three recent results on the two-dimensional nearest neighbour Potts model:
1. Strict convexity of equilibrium crystal shapes. (Campanino, I, Velenik).
2. Sharpness of phase transition (Beffara, Duminil-Copin).
3. Absence of non-translation invariant Gibbs states (Coquille, Duminil-Copin, I, Velenik).
All three are based on the stochastic geometric (random cluster) representation of the model.

Stephane Nonnenmacher: Resonances in quantum scattering

Our main object of interest is the scattering of a quantum particle (or a wave) on a localized potential, obstacle, or nontrivial metric, especially in cases where the corresponding ray dynamics admits trapped trajectories. Such a scattering system does not admit stationary modes of positive energies, but instead metastable modes associated with complex valued resonances. We are interested in the distribution of the long-living resonances (the most relevant ones from a physics point of view), which leads to an effective nonselfadjoint spectral problem.
In the high frequency/semiclassical régime, this distribution depends on the dynamical properties of the trapped rays. For instance, if the trapped rays form a "chaotic set" and are sufficiently unstable, one can prove the absence of resonances in a strip (a "resonance gap"). The determination of the optimal gap in this situation (which includes the case of hyperbolic surfaces of infinite area) remains an interesting open problem.
We will also consider the case where the trapped rays form a symplectic submanifold (of the phase space), carrying a normally hyperbolic flow. This situation is relevant for various applications, among which the dynamics of certain chemical reactions, or wave propagation on Kerr-de Sitter spacetimes. More surprisingly, this quantum scattering problem leads to a better understanding of the mixing properties of contact Anosov flows, a purely "classical dynamical" question.

Simone Warzel: The localization transition for random operators: recent surprises in the phase diagram on tree graphs

More than 50 years ago Anderson, Mott, Twose, and other physicists have proposed that the incorporation of a random potential in self adjoint operators of condensed matter physics results in a transition in the nature of the eigenstates from extended (e.g., plane waves) to localized, at least in certain energy ranges. The transition is accompanied in the reduction of conduction. As linear operators play key roles in many fields, myriads of other implications, and other interesting aspects (such as changes in the spectral gap statistics) have since then been noted of this transition. In this talk, I will give an introduction to the subject and describe recent progress in the understanding of the spectral and dynamical properties of such operators in case the underlying configuration space is a tree graph. Among the surprising phenomena which we discover is that even at weak disorder the regime of diffusive transport extends well beyond the one of the graph Laplacian into the regime of Lifshitz tails. As will be explained in the lecture, the mathematical mechanism for the appearance of conducting states in this non-perturbative regime are disorder-induced resonances.