Georgia Institute of Technology College of Sciences

It has been found about ten years ago that most of the real networks are not random ones in the Erdos-Renyi sense but have different topology (structure of the graph of interactions between the elements of a network). This finding generated a steady flux of papers analyzing structural aspects of networks. However, real networks are rather dynamical ones where the elements (cells, genes, agents, etc) are interacting dynamical systems. Recently a general approach to the studies of dynamical networks with arbitrary topology was developed. This approach is based on a symbolic dynamics and is in a sense similar to the one introduced by Sinai and the speaker for Lattice Dynamical Systems, where the graph of interactions is a lattice. The new approach allows to analyze a combined effect of all three features which characterize a dynamical network ( topology, dynamics of elements of the network and interactions between these elements) on its evolution. The networks are of the most general type, e.g. the local systems and interactions need not to be homogeneous, nor restrictions are imposed on a structure of the graph of interactions. Sufficient conditions on stability of dynamical networks are obtained. It is demonstrated that some subnetworks can evolve regularly while the others evolve chaotically. Some natural graph theoretical and dynamical questions appear in the farther developments of this approach. No preliminary knowledge of anything besides basic calculus and linear algebra is required to understand what is going on.

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.