Georgia Institute of Technology College of Sciences

Two-point symmetrizations are simple rearrangementsthat have been used to prove isoperimetric inequalitieson the sphere. For each unit vector u, there is atwo-point symmetrization that pushes mass towardsu across the normal hyperplane.How can full rotational symmetry be recovered from partialinformation? It is known that the reflections at d hyperplanes in general position generate a dense subgroup of O(d);in particular, a continuous function that is symmetric under thesereflections must be radial. How many two-point symmetrizationsare needed to verify that a function which increases under thesesymmetrizations is radial? I will show that d+1 such symmetrizationssuffice, and will discuss the ergodicity of the randomwalk generated by the corresponding folding maps on the sphere.(Joint work with G. R. Chambers and Anne Dranovski).

In this talk I will begin with our recent results on non-equilibrium steady states (NESS) of a microscopic heat conduction model, which is a stochastic particle system coupled to unequal heat baths. This stochastic model is derived from a mechanical chain model (Eckmann and Young 2006) by randomizing certain quantities while retaining the other features. We proved various results including the existence and uniqueness of NESS and the exponential rate of mixing. Then I will follow with an energy dependent Kac-type model that is obtained from an improved version of randomization of the “local" dynamics. We rigorously proved that this Kac-type model has a mixing rate $\sim t^{-2}$. In the end, I will show that slow (polynomial) mixing rates appear in a large class of statistical mechanics models.

It is an interesting well known fact that the relative entropy of the marginals of a density with respect to the Gaussian measure on Euclidean space satisfies a simple subadditivity property. Surprisingly enough, when one tries to achieve a similar result on the N-sphere a factor of 2 appears in the right hand side of the inequality (a result due to Carlen, Lieb and Loss), and this factor is sharp. Besides a deviation from the simple ``equivalence of ensembles principle'' in equilibrium Statistical Mechanics, this entropic inequality on the sphere has interesting ramifications in other fields, such as Kinetic Theory.In this talk we will present conditions on a density function on the sphere, under which we can get an ``almost'' subaditivity property; i.e. the factor 2 can be replaced with a factor that tends to 1 as the dimension of the sphere tends to infinity. The main tools for proving this result is an entropy conserving extension of the density from the sphere to Euclidean space together with a comparison of appropriate transportation distances such as the entropy, Fisher information and Wasserstein distance between the marginals of the original density and that of the extension. Time permitting, we will give an example that arises naturally in the investigation of the Kac Model.

Sources of single photons (as opposed to sources which produce on average a single photon) are of great current interest for quantum information processing. Perhaps surprisingly, it is not easy to produce a single photon efficiently and in a controlled way. Following earlier progress, recent experimental activity has resulted in the production of single photons by taking advantage of strong inter-particle interactions in cold atomic gases.I will show how the systematic use of the method of steepest descents can be used to understand the dynamics of the single photon source developed here at Georgia Tech and how this describes a kind of quantum scissors effect. In addition to the mathematical results, I will present the background quantum mechanics in a form suitable for a general audience. Joint work with Francesco Bariani and Paul Goldbart.