Georgia Institute of Technology College of Sciences

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.

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.

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.

Suppose that x(t) is a signal generated by a chaotic system and that the signal has been recorded in the interval [0,T]. We ask: What is the largest value t_f such that the signal can be predicted in the interval (T,T+t_f] using the history of the signal and nothing more? We show that the answer to this question is contained in a major result of modern information theory proved by Wyner, Ziv, Ornstein, and Weiss. All current algorithms for predicting chaotic series assume that if a pattern of events in some interval in the past is similar to the pattern of events leading up to the present moment, the pattern from the past can be used to predict the chaotic signal. Unfortunately, this intuitively reasonable idea is fundamentally deficient and all current predictors fall well short of the Wyner-Ziv bound. We explain why the current methods are deficient and develop some ideas for deriving an optimal predictor. [This talk is based on joint work with X. Liang and K. Serkh]. To view and/or participate in the webinar from wherever you are, click on:EVO.caltech.edu/evoNext/koala.jnlp?meeting=MvM2Ml2M2tDvDn9n9nDe9v