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Series: Math Physics Seminar

I will discuss local and nonlocal anisotropic heat transport along magnetic field lines in a tokamak, a device used to confine plasma undergoing fusion. I will give computational results that relate certain dynamical features of the magnetic field, e.g. resonance islands, chaotic regions, transport barriers, etc. to the asymptotic temperature profiles for heat transport along the magnetic field lines.

Series: Math Physics Seminar

We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices.Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially. This is joint work with Evans Harrell.

Series: Math Physics Seminar

We describe the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof. The talk is based on joint work with C. Hainzl, R. Seiringer and J. P. Solovej.

Series: Math Physics Seminar

We consider several models from solid state Physics and consider the problem offinding quasi-periodic solutions. We present a KAM theorem that showsthat given an approximate solution with good condition numbers, onecan find a true solution close by. The method of proof leads tovery efficient algorithms. Also it provides a criterion for breakdown.We will present the proof, the algorithms and some conjectures obtainedby computing in some cases. Much of the work was done with R. Calleja and X. Su.

Series: Math Physics Seminar

Hosted by Predrag Cvitanovic, School of Physics

Particle scattering processes at experiments such as the Large Hadron
Collider at CERN are described by scattering amplitudes. In quantum field
theory classes, students learn to calculate amplitudes using Feynman
diagram methods. This is a wonderful method for a process like electron +
positron -> muon^- + muon^+, but it is a highly challenging for a process
like gluon+gluon -> 5 gluons, which requires 149 diagrams even at the
leading order in perturbation theory. It turns out, however, that the
result for such gluon scattering processes is remarkably simple, in some
cases it is just a single term! This has lead to new methods for
calculating scattering amplitudes, and it has revealed that amplitudes
have
a surprisingly rich mathematical structure. The applications of these new
methods range from calculation of processes relevant for LHC physics to
theoretical explorations of quantum gravity. I will give a pedagogical
introduction to these new approaches to scattering theory and their
applications, not assuming any prior knowledge of quantum field theory or
Feynman rules.

Series: Math Physics Seminar

Shape optimization is the study of optimization problems whose
unknown is a domain in R^d. The seminar is focused on the understanding
of the case where admissible shapes are required to be convex. Such
problems arises in various field of applied mathematics, but also in
open questions of pure mathematics. We propose an analytical study of
the problem.
In the case of 2-dimensional shapes, we show some results for a large
class of functionals, involving geometric functionals, as well as
energies involving PDE. In particular, we give some conditions so that
solutions are polygons. We also give results in higher dimension,
concerned with the Mahler conjecture in convex geometry and the
Polya-Szego conjecture in potential theory. We particularly make the
link with the so-called Brunn-Minkowsky inequalities.

Series: Math Physics Seminar

The logarithmic Sobolev inequality (LSI) is a powerful tool for studying convergence to equilibrium in spin systems. The Bakry-Emery criterion implies LSI in the case of a convex Hamiltonian. What can be said in the nonconvex case? We present two recent sufficient conditions for LSI. The first is a Bakry-Emery-type criterion that requires only LSI (not convexity) for the single-site conditional measures. The second is a two-scale condition: An LSI on the microscopic scale (conditional measures) and an LSI on the macroscopic scale (marginal measure) are combined to prove a global LSI. We extend the two-scale method to derive an abstract theorem for convergence to the hydrodynamic limit which we then apply to the example of Guo-Papanicolaou-Varadhan. We also survey some new results.This work is joint with Grunewald, Otto, and Villani.

Series: Math Physics Seminar

We will show that a quantum xy-spin chain which is exposed to a randomexterior magnetic field satisfies a zero-velocity Lieb-Robinson bound. Thiscan be interpreted as dynamical localization for the spin chain or asabsence of information transport. We will also discuss a general result,which says that zero velocity LR-bounds in a quantum spin system implyexponential decay of ground state correlations. This is joint work withRobert Sims and Eman Hamza and motivated by recent works of Burrell-Osborneas well as Hastings.

Series: Math Physics Seminar

Hosted by Predrag Cvitanović, School of Physics, Georgia Tech.

Yves Couder and coworkers have recently reported the results of a startling
series of experiments in which droplets bouncing on a fluid surface exhibit
wave-particle duality and, as a consequence, several dynamical features
previously thought to be peculiar to the microscopic realm, including
single-particle diffraction, interference, tunneling and quantized orbits.
We explore this fluid system in light of the Madelung transformation,
whereby Schrodinger's equation is recast in a hydrodynamic form. Doing so
reveals a remarkable correspondence between bouncing droplets and subatomic
particles, and provides rationale for the observed macroscopic quantum
behaviour.
New experiments are presented, and indicate the potential value of
this hydrodynamic
approach to both visualizing and understanding quantum mechanics.

Series: Math Physics Seminar

I will talk about sequential decision making models based ondiffusion along heteroclinic networks of dynamical systems, i.e.,multiple saddle-type equilibrium points connected by heteroclinicorbits. The goal is to give a precise description of the asymptoticbehavior in the limit of vanishing noise.In particular, I will interpret exit times for stochastic dynamics asdecision making times and give a result on their asymptotic behavior.I will report on extensive data on decision making in no a priori biassetting obtained in a psychology experiment that I ran with JoshuaCorrell (University of Chicago),and compare the data with my theoretical results. I will also showthat the same kind of limiting distribution for exit times appears innonequilibrium models of statistical mechanics.