Georgia Institute of Technology College of Sciences

Orthogonal polynomials play a role in myriads of problems ranging from approximation theory to random matrices and signal processing. Generalizations of orthogonal polynomials - such as biorthogonal polynomials, cardinal series, Muntz polynomials, are used for example, in number theory and numerical analysis. We discuss some of these, and some potential research projects involving them.

The first part of this work deals with open dynamical systems. A natural question of how the survival probability depends upon a position of a hole was seemingly never addresses in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period), which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole. In the second part we consider some classes of cellular automata called Deterministic Walks in Random Environments on \mathbb Z^1. At first we deal with the system with constant rigidity and Markovian distribution of scatterers on \mathbb Z^1. It is shown that these systems have essentially the same properties as DWRE on \mathbb Z^1 with constant rigidity and independently distributed scatterers. Lastly, we consider a system with non-constant rigidity (so called process of aging) and independent distribution of scatterers. Asymptotic laws for the dynamics of perturbations propagating in such environments with aging are obtained.

It is still not known whether every genus g Lefschetz fibration over the 2-sphere admits a section or not. In this talk, we will give a brief background information on Lefschetz fibrations and talk about sections of genus two Lefschetz fibration. We will observe that any holomorphic genus two Lefschetz fibration without seperating singular fibers admits a section. This talk is accessible to anyone interested in topology and geometry.

In this presentation, interactions between spectra of classical Gaussian ensembles and subsequence problems are studied with the help of the powerful machinery of Young tableaux. For the random word problem, from an ordered finite alphabet, the shape of the associated Young tableaux is shown to converge to the spectrum of the (generalized) traceless GUE. Various properties of the (generalized) traceless GUE are established, such as a law of large number for the extreme eigenvalues and the convergence of the spectral measure towards the semicircle law. The limiting shape of the whole tableau is also obtained as a Brownian functional. The Poissonized word problem is finally talked, and, with it, the convergence of the whole Poissonized tableaux is derived.