Georgia Institute of Technology College of Sciences

Orthogonal Polynomials play a key role in analysis of random matrices. We discuss universality limits in the so-called unitary case, showing how the universality limit reduces to an asymptotic involving reproducing kernels associated with orthogonal polynomials. As a consequence, we show that universality holds in measure for any compactly supported measure.

Consider a nonlinear Schrodinger equation in $R^3$ whose linear part has three or more eigenvalues satisfying some resonance conditions. Solutions which are initially small in $H^1 \cap L^1(R^3)$ and inside a neighborhood of the first excited state family are shown to converge to either a first excited state or a ground state at time infinity. An essential part of our analysis is on the linear and nonlinear estimates near nonlinear excited states, around which the linearized operators have eigenvalues with nonzero real parts and their corresponding eigenfunctions are not uniformly localized in space. This is a joint work with Kenji Nakanishi and Tuoc Van Phan.The preprint of the talk is available at http://arxiv.org/abs/1008.3581

The study of Legendrian and transversal knots has been an essential part of contact topology for quite some time now, but until recently their study in overtwisted contact structures has been virtually ignored. In the past few years that has changed. I will review what is know about such knots and discuss recent work on the "geography" and "botany" problem.

Lopsided bipartite graphs naturally appear in advertising setting. One side is all the eyeballs and the other side is all the advertisers. An edge is when an advertiser wants to reach an eyeball, aka, ad targeting. Such a bipartite graph is lopsided because there are only a small number of advertisers but a large number of eyeballs. We give algorithms which have running time proportional to the size of the smaller side, i.e., the number of advertisers. One of the main ideas behind our algorithm and as well as the analysis is a property, which we call, monotonic quality bounds. Our algorithm is flexible as it could easily be adapted for different kinds of objective functions. Towards the end of the talk we will describe a new matching polytope. We show that our matching polytope is not only a new linear program describing the classical matching polytope, but is a new polytope together with a new linear program. This part of the talk is still theoretical as we only know how to solve the new linear program via an ellipsoid algorithm. One feature of the polytope, besides being intriguing, is that it has some notion of fairness built in. This is important for advertising since if an advertiser wants to reach 10 million users of type A or type B, advertiser won't necessarily be happy if we show the ad to 10 million users of type A only (though it fulfills the advertising contract in a technical sense).