Prize Lectures

Fractal uncertainty principle and quantum chaos

Fractal uncertainty principle states that no function can be localized close to a fractal set simultaneously in position and momentum. The strongest version so far has been obtained in one dimension by Bourgain and the speaker with recent higher dimensional advances by Han and Schlag. It has applications to spectral gaps in chaotic scattering and to localization and control of high energy eigenfunctions.

More precisely, using the work with Bourgain and earlier work with Zahl, Jin and the speaker proved that on hyperbolic surfaces, the mass of an eigenfunction on an open set is bounded from below independently of the energy. As shown by Jin, these results lead to observability and control for the Schrödinger equation. Another application is the existence of a spectral gap for every convex co-compact hyperbolic surface, which implies local energy decay of waves at high frequency.

Universality in numerical computation with random data: Case studies and analytical results.

The speaker will discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations, rigorous results, and some speculations about computation in a broader sense.

Joint with C. Pfrang, G. Menon, S. Olver  and Thomas Trogdon.

Telegraph equation from the six-vertex model

I will explain how a second order hyperbolic PDE, known as the telegraph equation, arises in the asymptotic of the height function of a celebrated model of 2d statistical mechanics - the six-vertex (or square ice) model. Homogeneous equation describes the limit shape in the system, and its stochastic inhomogeneous version governs the fluctuations.

Reversibility, Irreversibility, Friction and Nonequilibrium Ensembles in Navier-Stokes equations

A proposal for a theory of equivalent ensembles in nonequilibrium statistical mechanics.

The case of Navier-Stokes fluids. Equivalent irreversible and reversible friction for incompressible, periodic boundary conditions, constant forcing flows in 2/3D.

The prize recipients for 2014, 2015, and 2016 will be formally recognized during this brief ceremony. 

The associated lectures for each of the prizes are separately scheduled.

Some recent progress on mean field spin glasses.

Spin glass models were initially introduced by theoretical physicists in order to study some strange magnetic behavior of certain alloys. They exhibit several features, such as quenched disorder and frustration, that are commonly shared in many real world problems involving randomized combinatorial optimizations. In this talk, we will discuss some recent progress on the famous Sherrington-Kirkpatrick mean field spin glass as well as its variants. We will explain Parisi's solution to the computation of the ground state energies and give descriptions on the energy landscapes of the models in relation to the computational hardness of the ground state energies.

Nonlinear Gibbs measure and equilibrium Bose gases

Nonlinear Gibbs measure plays an important role in the studies of  Euclidean Quantum Field Theory, nonlinear PDEs and stochastic PDEs. I will discuss the natural emergence of this measure as the mean-field limit of bosonic equilibrium states. The talk is based on joint work with Mathieu Lewin and Nicolas Rougerie.

Emergent structures in statistical mechanics

Equilibrium states of classical and quantum systems can often be understood in terms of spontaneously emergent random geometric structures.   Examples of such can be seen in the spontaneous organization of Ising and Potts spins into cliques,  whose statistics are given by the Fortuin - Kasteleyn random cluster models, the random current representation of the equilibrium Gibbs states of Ising / phi^4 models,  and loop based organization of certain quantum spin chains into spin zero clusters.  Uncovering the hidden stochastic geometric features allows insights on the model’s phase structure, and the nature of the correlation functions.  For quantum spin chains the loop representations are of relevance for the phenomena of dimerization, the emergence of boundary spin excitations, and in some cases of topological states.  Mentioned in the talk will be results based on both old and recent  collaborations. 

Schrödinger operators with thin spectra

In the early days of the spectral analysis of Schrödinger operators, the essential spectrum commonly turned out to be an interval, or at least a union of intervals. In the 1980's, new types of spectra were discovered, displaying a Cantor set structure. In this talk we present recent work on Schrödinger operators whose spectra are even thinner, having zero Lebesgue measure, or even zero Hausdorff dimension. The mechanisms leading to these results will be elucidated. This is joint work with Jake Fillman, Anton Gorodetski and Milivoje Lukic.

Article: Damanik, D., Fillman, J. & Gorodetski, A. Ann. Henri Poincaré (2014) 15: 1123. https://doi.org/10.1007/s00023-013-0264-6

Instability of pre-existing resonances in the DC Stark effect

This talk is based on the article:

I. Herbst, J. Rama, "Instability of Pre-Existing Resonances Under a Small Constant Electric Field", Ann. Henri Poincar$\acute{\rm e}$ 16 (2015), 2783 - 2835.

A simple model operator (Friedrichs model) with a pre-existing resonance $r_0$, defined in a dilation or translation analytic framework, is considered. Adding a potential corresponding to a small constant electric field of strength $f>0$ (DC Stark effect) perturbs the pre-existing resonance into (infinitely many) resonances of the resulting operator. It is shown that the pre-existing resonance $r_0$ is unstable in the weak field limit $f\downarrow 0$, in the sense that the new resonances do not converge to $r_0$ as $f\downarrow 0$. In a dilation analytic framework this instability result in the DC Stark effect is in contrast to the AC Stark effect (time-periodic electric field), where the pre-existing resonance is stable in the weak field limit.

Article: Herbst, I. & Rama, J. Ann. Henri Poincaré (2015) 16: 2783. https://doi.org/10.1007/s00023-014-0389-2

Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems

We consider translation invariant gapped quantum spin systems satisfying the Lieb-Robinson bound and containing single-particle states in a ground state representation. Following the Haag-Ruelle approach from relativistic quantum field theory, we construct states describing collisions of several particles, and define the corresponding S-matrix. For the purpose of our analysis we adapt the concepts of almost local observables and energy-momentum transfer (or Arveson spectrum) from relativistic QFT to the lattice setting. Our results hold, in particular, in the Ising model in strong transverse magnetic fields. Based on these developments, a general discussion of the problem of asymptotic completeness for gapped quantum spin systems will be given.

Based on joint work with S. Bachmann and P. Naaijkens.

Article: Bachmann, S., Dybalski, W. & Naaijkens, P. Ann. Henri Poincaré (2016) 17: 1737. https://doi.org/10.1007/s00023-015-0440-y