Georgia Institute of Technology College of Sciences

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.

I will discuss the fluctuations of the spectral density for the Wigner ensemble on the optimal scale. We study the fluctuations of the Stieltjes transform, and improve the known bounds on the optimal scale. As an application, we derive the semicircle law at the edge of the spectrum. This is joint work with Claudio Cacciapuoti and Benjamin Schlein.

The (blobbed) topological recursion is a recursive structure which defines, for any initial datagiven by symmetric holomorphic 1-form \phi_{0,1}(z) and 2-form \phi_{0,2}(z_1,z_2) (and symmetricn-forms \phi_{g,n} for n >=1 and g >=0), a sequence of symmetric meromorphic n-forms\omega_{g,n}(z_1,...,z_n) by a recursive formula on 2g - 2 + n.If we choose the initial data in various ways, \omega_{g,n} computes interesting quantities. A mainexample of application is that this topological recursion computes the asymptotic expansion ofhermitian matrix integrals. In this talk, matrix models with also serve as an illustration of thisgeneral structure.