Georgia Institute of Technology College of Sciences

In 1966 V. Arnold observed that solutions to the Euler equations of incompressible fluids can be viewed as geodesics of the kinetic energy metric on the group of volume-preserving diffeomorphisms. This introduced Riemannian geometric methods into the study of ideal fluids. I will first review this approach and then describe results on the structure of singularities of the associated exponential map and (time premitting) related recent developments.

In this talk I will discuss bounds on the slice genus of aknot coming from it's representation as a braid closure, starting withthe slice-Bennequin inequality. From there I will use surfacebraiding techniques of Rudolph and Kamada to exhibit a new lower boundon the ribbon genus of a knot, given some knowledge about what slicesurfaces it bounds.

We introduce a new model for cell phone signal problem, which is stochastic van der Pol oscillator with condition that ensures global boundedness in phase space and keeps unboundedness for frequency. Also we give a new definition for stochastic Poincare map and find a new approximation to return time and point. The new definition is based on the numerical observation. Also we develop a new approach by using dynamic tools, such as method of averaging and relaxation method, to estimate the return time and return point. Thus we can show that the return time is always not Gaussian and return point's distribution is not symmetric under certain section.

Motivated by rich applications in science and engineering, I am interested in controlling systems that are characterized by multiple scales, geometric structures, and randomness. This talk will focus on my first two steps towards this goal. The first step is to be able to simulate these systems. We developed integrators that do not resolve fast scales in these systems but still capture their effective contributions. These integrators require no identification of underlying slow variables or processes, and therefore work for a broad spectrum of systems (including stiff ODEs, SDEs and PDEs). They also numerically preserve intrinsic geometric structures (e.g., symplecticity, invariant distribution, and other conservation laws), and this leads to improved long time accuracy. The second step is to understand what noises can do and utilize them. We quantify noise-induced transitions by optimizing probabilities given by Freidlin-Wentzell large deviation theory. In gradient systems, transitions between metastable states were known to cross saddle points. We investigate nongradient systems, and show transitions may instead cross unstable periodic orbits. Numerical tools for identifying periodic orbits and for computing transition paths are proposed. I will also describe how these results help design control strategies.