Georgia Institute of Technology College of Sciences

In large part, the waves that we observe in the open ocean are created by wind blowing over the water. The precise nature of this process occurs has been intensely studied, but is still not understood very well at a mathematically rigorous level. In this talk, we side-step that issue, somewhat, and consider the steady problem. That is, we prove the existence of small-amplitude traveling waves in a two phase air-water system that can be viewed as the eventual product of wind generation. This is joint work with Oliver Buhler and Jalal Shatah.

I will present two "real life" applications of Riemannian geometry,one in the field of continuum mechanics, another in the field ofmachine learning (nonlinear dimensionality reduction). I will providequick introductions in order to make the talk accessible to anaudience with no background in either field.

This talk will be an amalgamation of aspects of scientific computing - development, verification, and interpretation - with application to the Vlasov-Poisson (VP) system, an important nonlinear partial differential equation containing the essential difficulties of collisionless kinetic theories. I will describe our development of a discontinuous Galerkin (DG) algorithm, its verification via convergence studies and comparison to known Vlasov results, and our interpretation of computational results in terms of dynamical systems ideas. The DG method was invented for solving a neutron transport model, successfully adapted to fluid motion including shock propagation, applied to the Boltzmann equation, and developed in the general context of conservation laws, and elliptic and parabolic equations. Our development for the VP system required the simultaneous approximation of the hyperbolic Vlasov equation with the elliptic Poisson equation, which created new challenges. I will briefy discuss advantages of the method and describe our error estimates and recurrence calculations for polynomial bases. Then, I will show results from a collection of benchmark computations of electron plasma dynamics, including i) convergence studies of high resolution linear and nonlinear Landau damping with a comparison to theoretical parameter dependencies ii) the nonlinear two-stream instability integrated out to (weak) saturation into an apparently stable equilibrium (BGK) state with detailed modeling of this state, and iii) an electric field driven (dynamically accessible) example that appears to saturate into various periodic solutions. I will interpret such final states, in analogy to finite-dimensional Hamiltonian theory, as Moser-Weinstein periodic orbits, and suggest a possible variational path for proof of their existence. Finally, I will comment briefly on recent progress on extensions to the Maxwell-Vlasov system, including estimates and computational results.

The heart is an excitable system in which electrical waves normally propagate in a coordinated manner to produce an effective mechanical contraction. Rapid pacing can lead to the development of alternans, a period-doubling bifurcation in electrical response in which successive beats have long and short responses despite a constant pacing period. Alternans can develop into higher-order rhythms as well as spatiotemporally complex patterns that reflect large regions of dispersion in electrical response. These states disrupt synchrony and compromise the heart's mechanical function; indeed, alternans has been observed clinically as a precursor to dangerous arrhythmias, including ventricular fibrillation. In this talk, we will show experimental examples of alternans, describe how alternans develops using a mathematical and computational approach, and discuss the nonlinear dynamics of several possible mechanisms for alternans as well as the conditions under which they are likely to be important in initiating dangerous cardiac arrhythmias.