Seminars and Colloquia by Series

Thursday, October 29, 2015 - 17:00 , Location: Skiles 006 , Mikel Viana , Georgia Tech (Math) , Organizer:
We consider fibered holomorphic dynamics, generated by a skew product over an irrational translation of the torus. The invariant object that organizes the dynamics is an invariant torus. Often one can find an  approximately invariant torus K_0, and we construct an invariant torus,  starting from K_0. The main technique is a KAM iteration in a-posteriori format. In this talk we give the details of the iterative procedure using the geometric and number-theoretic conditions presented last time.
Tuesday, October 6, 2015 - 17:00 , Location: Skiles 005 , Mikel de Viana , Georgia Tech , Organizer:
We consider fibered holomorphic dynamics, generated by a skew product over an irrational translation of the torus. The invariant object that organizes the dynamics is an invariant torus. Often one can find an  approximately invariant torus K_0, and we construct an invariant torus,  starting from K_0. The main technique is a KAM iteration in a-posteriori format. The asymptotic properties of the derivative cocycle A_K  play a crucial role: In this first talk we will find suitable geometric and number-theoretic conditions for A_K. Later, we will see how to relax these conditions.
Tuesday, September 29, 2015 - 17:00 , Location: Skiles 005 , Rafael de la Llave , Georgia Tech (Math) , Organizer:
We will review the notion of Whitney differentiability and the Whitney embedding theorem.  Then, we will also review its applications in KAM theory (continuation of last week's talk).
Tuesday, September 22, 2015 - 17:00 , Location: Skiles 254 , Rafael de la Llave , Georgia Institute of Technology , Organizer: Lei Zhang
We will review the notion of Whitney differentiability and the Whitney embedding theorem.  Then, we will also review its applications in KAM theory. 
Tuesday, September 15, 2015 - 17:00 , Location: Skiles 005 , Xiaolong He , Georgia Tech (Math)/Hunan University , Organizer:
We investigate the existence of quasi-periodic solutions for state-dependent delay differential equationsusing the parameterization method, which is different from the usual way-working on the solution manifold.  Under the assumption of finite-time differentiability of functions and exponential dichotomy, the existence and smoothness of quasi-periodic solutions are investigated by using contraction arguments We also develop a  KAM theory  to seek analytic quasi-periodic solutions. In contrast with the finite differentonable theory, this requires adjusting parameters. We prove that the set of parameters which guarantee the existence of analytic quasi-periodic solutions is of positive measure. All of these results are given in an a-posterior form. Namely, given a approximate solution satisfying some non-degeneracy conditions, there is a true solution nearby.
Tuesday, September 8, 2015 - 17:00 , Location: Skiles 005 , Jiaqi Yang , Georgia Tech , Organizer:
(continuation of last week's seminar): We will discuss KAM results for symplectic and presymplectic maps. Firstly, we will study geometric properties of a symplectic dynamical system which will allow us to prove a KAM theorem in a-posteriori format. Then, a corresponding theorem for a parametric family of symplectic maps will be presented. Finally, using similar method, we will extend the theorems to presymplectic maps. These results appear in the work of Alishah, de la Llave, Gonzalez, Jorba and Villanueva.
Tuesday, September 1, 2015 - 17:00 , Location: Skiles 005 , Jiaqi Yang , Georgia Tech , Organizer:
We will discuss KAM results for symplectic and presymplectic maps. Firstly, we will study geometric properties of a symplectic dynamical system which will allow us to prove a KAM theorem in a-posteriori format. Then, a corresponding theorem for a parametric family of symplectic maps will be presented. Finally, using similar method, we will extend the theorems to presymplectic maps. These results appear in the work of Alishah, de la Llave, Gonzalez, Jorba and Villanueva.
Wednesday, April 16, 2014 - 12:05 , Location: Skiles , Rafael de la Llave , Georgia Tech , Organizer:
We prove Ruelle's Entropy Inequality for C^1 maps. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.
Friday, April 11, 2014 - 13:05 , Location: Skiles 005 , Mikel J. de Viana , Georgia Tech , Organizer:
We finish our discussion on  Oseledets Theorem by proving the convergence of the filtration. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.
Friday, April 4, 2014 - 13:05 , Location: Skiles 005 , Mikel J. de Viana , Georgia Tech , Organizer:
We begin the proof of Oseledets Theorem. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.

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