Seminars and Colloquia by Series

Tuesday, October 8, 2013 - 16:05 , Location: skiles 006 , Mikel J. de Viana , Georgia Tech , Organizer:
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.
Tuesday, March 26, 2013 - 16:30 , Location: Skiles 006 , Xifeng Su , Academy of Mathematics and Systems Science, Chinese Academy of Sciences , , Organizer:
We consider the evolutionary first order nonlinear partial differential equations of the most general form \frac{\partial u}{\partial t} + H(x, u, d_x u)=0.By virtue of introducing a new type of solution semigroup, we  establish the weak KAM theorem for such partial differential equations, i.e. the existence of weak KAM solutions or viscosity solutions. Indeed, by employing dynamical approach for characteristics, we develop the theory of associated global viscosity solutions in general.  Moreover, the solution semigroup acting on any given continuous function will converge to a uniform limit as the time goes to infinity. As an application, we prove that such limit satisfies the  the associated stationary first order  partial differential equations:  H(x, u, d_x u)=0.
Tuesday, February 26, 2013 - 16:30 , Location: Skiles 06 , F. Fenton , Georgia Tech (Physics) , Organizer: Rafael de la Llave
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
Tuesday, January 29, 2013 - 16:30 , Location: Skiles 06 , Rafael de la Llave , Georgia Tech , Organizer: Rafael de la Llave
We will present the method of correctly aligned windows and show how it can lead to large scale motions when there are homoclinic orbits to a normally hyperbolic manifold.
Tuesday, November 20, 2012 - 16:30 , Location: Skiles 006 , Rafael de la Llave , Georgia Tech , Organizer: Rafael de la Llave
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.
Tuesday, November 13, 2012 - 16:35 , Location: Skiles 006 , Mikel J. de Viana , Georgia Tech , Organizer:
Thursday, November 8, 2012 - 16:30 , Location: Skiles 06 , Rafael de la Llave , Georgia Tech , Organizer: Rafael de la Llave
The existence of several objects in dynamics can be reduced to the existence of solutions of several functional equations, which then, are dealt with using fixed point theorems (e.g. the contraction mapping principle). This opens the possibility to take numerical approximations and validate them. This requires to take into account truncation, roundoff and other sources of error. I will try to present the principles involved as well as some practical implementations of a basic library. Much of this is work with others including D. Rana, R. Calleja, J. L. Figueras.
Tuesday, November 6, 2012 - 16:35 , Location: Skiles 006 , Mikel J. de Viana , Georgia Tech , Organizer:
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.  A more complicated problem has been the study of the action by left multiplication by positive diagonal matrices, A. We will discuss the main ideas in the work of Einsiedler, Katok and Lindenstrauss where a measure classification is obtained, assuming that there is a one parameter subgroup of A which acts with positive entropy. The first talk is devoted to completing our understanding of the unipotent actions in SL(2,\Z)\ SL(2,\R), a la Ratner, because it is essential to understanding the "low entropy method" of Lindenstrauss. We will then introduce the necessary tools and assumptions, and next week we will complete the classification by application of two complementary methods.
Thursday, November 1, 2012 - 16:30 , Location: Skiles 06 , Rafael de la Llave , Georgia Tech , Organizer: Rafael de la Llave
"Shadowing" in dynamical systems is the property that an approximate orbit (satisfying some additional properties) can be followed closely by a true orbit. This is  a basic tool to construct complicated orbits since construction of approximate orbits is sometimes easier. It is also important in applications since numerical computations produce only approximate orbits and it requires an extra argument to show that the approximate ofbit produced by the computer corresponds to a real orbit. There are three standard mechanicsms for shadowing: Hyperbolicity, topological methods, shadowing of minimizers. We will present hyperbolicity.
Tuesday, October 9, 2012 - 16:30 , Location: Skiles 06 , Maciej Capinski , Georgia Tech , Organizer:
In the talk we will present a mechanism of diffusion in the Planar Circular Restricted Three Body Problem. The mechanism is similar to the one that appeared in the celebrated work of V. I. Arnold [Dokl. Akad. Nauk SSSR 156 (1964), 9–12]. Arnold conjectured that this phenomenon, usually called Arnold diffusion, appears in the three body problem. The presented method is a step towards a proof of the conjecture. In this second, and final part of the talk, we discuss how to prove transversal intersections of invariant manifolds in the circular problem and how these lead to diffusion in the elliptic problem.