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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.

Friday, March 28, 2014 - 13:05 ,
Location: Skiles 005 ,
Lei Zhang ,
Georgia Tech ,
Organizer:

We present the proof of the Shannon-McMillan-Breiman 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.

Friday, March 14, 2014 - 13:05 ,
Location: Skiles 005 ,
Lei Zhang ,
Georgia Institute of Technology ,
Organizer:

We introduce concepts of entropy and methods of calculation of entropy and examples. 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, March 7, 2014 - 13:00 ,
Location: Skiles 05 ,
Mikel J. de Viana ,
Georgia Tech ,
Organizer: Rafael de la Llave

We will present a proof of the subadditive ergodic 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.

Friday, February 28, 2014 - 13:00 ,
Location: Skiles 005 ,
Lei Zhang ,
Georgia Tech ,
lzhang98@math.gatech.edu ,
Organizer: Lei Zhang

This is a reading seminar on smooth ergodic theory. In the first talk we will introduce some basic notions of ergodic theory and proof Birkhoff Ergodic Theorem.