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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.

Towards the proof of diffusion in the Jupiter-Sun restricted three body problem (second, final part)

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.

Tuesday, October 2, 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.

Wednesday, September 19, 2012 - 16:00 ,
Location: Skiles 06 ,
Lei Zhang ,
Georgia Tech ,
Organizer: Rafael de la Llave

Continuation of the exposition of N. Fenichel classical paper on the persistence of Normally Hyperbolic invariant manifolds.

Wednesday, September 12, 2012 - 16:00 ,
Location: Skiles 06 ,
Lei Zhang ,
Georgia Institute of technology ,
Organizer: Rafael de la Llave

We will present the classical work of N. Fenichel on persitence of overflowing manifolds.

Wednesday, September 5, 2012 - 16:00 ,
Location: Skiles 06 ,
Rafael de la Llave ,
Georgia Institute of technology ,
Organizer: Rafael de la Llave

We will present a classical proof of the center stable manifold.