Georgia Institute of Technology College of Sciences

It is well-known that a deterministic dynamical system can exhibit stochastic behavior that is due to the fact that instability along typical trajectories of the system drives orbits apart, while compactness of the phase space forces them back together. The consequent unending dispersal and return of nearby trajectories is one of the hallmarks of chaos. The hyperbolic theory of dynamical systems provides a mathematical foundation for the paradigm that is widely known as "deterministic chaos" -- the appearance of irregular chaotic motions in purely deterministic dynamical systems. This phenomenon is considered as one of the most fundamental discoveries in the theory of dynamical systems in the second part of the last century. The hyperbolic behavior can be interpreted in various ways and the weakest one is associated with dynamical systems with non-zero Lyapunov exponents. I will discuss the still-open problem of whether dynamical systems with non-zero Lyapunov exponents are typical. I will outline some recent results in this direction. The genericity problem is closely related to two other important problems in dynamics on whether systems with nonzero Lyapunov exponents exist on any phase space and whether nonzero exponents can coexist with zero exponents in a robust way.

We show that the LIA approximation of the incompressible Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for higher-dimensional vortex filaments and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively. This framework, in particular, allows one to define the symplectic structures on the spaces of vortex sheets.

Thresholds for increasing properties are a central concern in probabilistic combinatorics and elsewhere. (An increasing property, say F, is a superset-closed family of subsets of some (here finite) set X; the threshold question for such an F asks, roughly, about how many random elements of X should one choose to make it likely that the resulting set lies in F? For example: about how many random edges from the complete graph K_n are typically required to produce a Hamiltonian cycle?) We'll discuss recent progress and lack thereof on a few threshold-type questions, and try to say something about a ludicrously general conjecture of G. Kalai and the speaker to the effect that there is always a pretty good naive explanation for a threshold being what it is.