Georgia Institute of Technology College of Sciences

Hamiltonian systems typically exhibit a mixture of chaos and regularity, complicating any scheme to partition phase space and extract a symbolic description of the dynamics. In particular, the dynamics in the vicinity of stable islands can exhibit extremely complicated topology. We present an approach to extracting symbolic dynamics in such systems using networks of nested heteroclinic tangles-- fundamental geometric objects that organize phase space transport. These tangles can be used to progressively approximate the behavior in the vicinity of stable island chains. The net result is a symbolic approximation to the dynamics, and an associated phase-space partition, that includes the influence of stable islands. The utility of this approach is illustrated by examining two applications in atomic physics -- the chaotic escape of ultracold atoms from an atomic trap and the chaotic ionization of atoms in external fields.

Large-scale inverse problems arise in a variety of importantapplications in image processing, and efficient regularization methodsare needed to compute meaningful solutions. Much progress has beenmade in the field of large-scale inverse problems, but many challengesstill remain for future research. In this talk we describe threecommon mathematical models including a linear, a separable nonlinear,and a general nonlinear model. Techniques for regularization andlarge-scale implementations are considered, with particular focusgiven to algorithms and computations that can exploit structure in theproblem. Examples will illustrate the properties of these algorithms.

The map coloring problem is one of the major catalysts of the tremendous development of graph theory. It was observed by Tutte that the problem of the face-coloring of an planar graph can be formulated in terms of integer flows of the graph. Since then the topic of integer flow has been one of the most attractive in graph theory. Tutte had three famous fascinating flow conjectures: the 3-flow conjecture, the 4-flow conjecture and the 5-flow conjecture. There are some partial results for these three conjectures. But in general, all these 3 conjectures are open. Group connectivity is a generalization of integer flow of graphs. It provides us with contractible flow configurations which play an important role in reducing the graph size for integer flow problems, it is also related to all generalized Tutte orientations of graphs. In this talk, I will present an introduction and survey on group connectivity of graphs as well as some open problems in this field.

For every quantum group one can define two invariants of 3-manifolds:the WRT invariant and the Hennings invariant. We will show that theseinvariants are equivalentfor quantum sl_2 when restricted to the rational homology 3-spheres.This relation can be used to solve the integrality problem of the WRT invariant.We will also show that the Hennings invariant produces integral TQFTsin a more natural way than the WRT invariant.