Georgia Institute of Technology College of Sciences

Incompressible Euler equation is known to be the geodesic flow on the manifold of volume preserving maps. In this informal seminar, we will discuss how this geometric and Lagrangian point of view may help us understand certain analytic and dynamic aspects of this PDE.

Given f: \C \times T^1 to itself, an analytic perturbation of a fibered rotation map , we will present two proofs of existence of an analytic conjugation of f to the fibered rotation , on a neighborhood of {0} \times T^1. This talk will be self- contained except for some usual "tricks" from KAM theory and which will be explained better in another talk. In the talk we will discuss carefully the number theoretic conditions on the fibered rotation needed to obtain the theorem.

The heart is an electro-mechanical system in which, under normal conditions, electrical waves propagate in a coordinated manner to initiate an efficient contraction. In pathologic states, propagation can destabilize and exhibit period-doubling bifurcations that can result in both quasiperiodic and spatiotemporally chaotic oscillations. In turn, these oscillations can lead to single or multiple rapidly rotating spiral or scroll waves that generate complex spatiotemporal patterns of activation that inhibit contraction and can be lethal if untreated. Despite much study, little is known about the actual mechanisms that initiate, perpetuate, and terminate reentrant waves in cardiac tissue. In this talk, I will discuss experimental and theoretical approaches to understanding the dynamics of cardiac arrhythmias. Then I will show how state-of-the-art voltage-sensitive fluorescent dyes can be used to image the electrical waves present in cardiac tissue, leading to new insights about their underlying dynamics. I will establish a relationship between the response of cardiac tissue to an electric field and the spatial distribution of heterogeneities in the scale-free coronary vascular structure. I will discuss how in response to a pulsed electric field E, these heterogeneities serve as nucleation sites for the generation of intramural electrical waves with a source density ?(E) and a characteristic time constant ? for tissue excitation that obeys a power law. These intramural wave sources permit targeting of electrical turbulence near the cores of the vortices of electrical activity that drive complex fibrillatory dynamics. Therefore, rapid synchronization of cardiac tissue and termination of fibrillation can be achieved with a series of low-energy pulses. I will finish with results showing the efficacy and clinical application of this novel low energy mechanism in vitro and in vivo. e

We will continue the presentation of some tools to obtain computer assisted proofs in dynamics using functional analysis methods. Joint work with D. Rana, R. Calleja.