Georgia Institute of Technology College of Sciences

The study of actions of subgroups of SL(k,\R) on the space of unimodular lattices in \R^k has received considerable attention since at least the 1970s. The dynamical properties of these systems often have important consequences, such as for equidistribution results in number theory. In particular, in 1984, Margulis proved the Oppenheim conjecture on values of indefinite, irrational quadratic forms by studying one dimensional orbits of unipotent flows.

Bio-polymerization processes like transcription and translation are central to a proper function of a cell. The speed at which the bio-polymer grows is affected both by number of pauses of elongation machinery, as well their numbers due to crowding effects. In order to quantify these effects in fast transcribing ribosome genes, we rigorously show that a classical traffic flow model is a limit of mean occupancy ODE model.

We will present a classical proof of the center stable manifold.

Hyperbolic actions of Z^k and R^k arise naturally in algebraic and geometric context. Algebraic examples include actions by commuting automorphisms of tori or nilmanifolds and, more generally, affine and homogeneous actions on cosets of Lie groups. In contrast to hyperbolic actions of Z and R, i.e. Anosov diffeomorphisms and flows, higher rank actions exhibit remarkable rigidity properties, such as scarcity of invariant measures and smooth conjugacy to a small perturbation. I will give an overview of results in this area and discuss recent progress.

A linear cocycle over a diffeomorphism f of a manifold M is an automorphism of a vector bundle over M that projects to f. An important example is given by the differential Df or its restriction to an invariant sub-bundle. We consider a Holder continuous linear cocycle over a hyperbolic system and explore what conclusions can be made based on its properties at the periodic points of f. In particular, we obtain criteria for a cocycle to be isometric or conformal and discuss applications and further developments.

Consider a hyperbolic basic set of a smooth diffeomorphism. We are interested in the transitivity of Holder skew-extensions with fiber a non-compact connected Lie group. In the case of compact fibers, the transitive extensions contain an open and dense set. For the non-compact case, we conjectured that this is still true within the set of extensions that avoid the obvious obstructions to transitivity. Within this class of cocycles, we proved generic transitivity for extensions with fiber the special Euclidean group SE(2n+1) (the case

The discrete Schrodinger operator with Fibonacci potential is a central model in the study of electronic properties of one-dimensional quasicrystals. Certain renormalization procedure allows to reduce many questions on specral properties of this operator to the questions on dynamical properties of a so called trace map. It turnes out that the trace map is hyperbolic, and its measure of maximal entropy is directly related to the integrated density of states of the Fibonacci Hamiltonian.

The study of transport is an active area of applied mathematics of interest to fluid mechanics, plasma physics, geophysics, engineering, and biology among other areas. A considerable amount of work has been done in the context of diffusion models in which, according to the Fourier-­‐Fick’s prescription, the flux is assumed to depend on the instantaneous, local spatial gradient of the transported field.

We present a novel method to find KAM tori in degenerate (nontwist) cases. We also require that the tori thus constructed have a singular Birkhoff normal form. The method provides a natural classification of KAM tori which is based on Singularity Theory.The method also leads to effective algorithms of computation, and we present some preliminary numerical results. This work is in collaboration with R. de la Llave and A. Gonzalez.

We generalize some notions that have played an important role in dynamics, namely invariant manifolds, to the more general context of difference equations. In particular, we study Lagrangian systems in discrete time. We define invariant manifolds, even if the corresponding difference equations can not be transformed in a dynamical system. The results apply to several examples in the Physics literature: the Frenkel-Kontorova model with long-range interactions and the Heisenberg model of spin chains with a perturbation. We use a modification of the parametrization