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Friday, March 11, 2016 - 13:05 ,
Location: Skiles 170 ,
Hongyu Cheng ,
Georgia Tech ,
Organizer: Lei Zhang

We present some basic results from the theory of stochastic processes and investigate the properties of some standard continuous-time stochastic processes. Firstly, we give the deﬁnition of a stochastic process. Secondly, we introduce Brownian motion and study some of its properties. Thirdly, we give some classical examples of stochastic processes in continuous time and at last prove some famous theorems.

Friday, March 4, 2016 - 13:05 ,
Location: Skiles 170 ,
Jiayin Jin ,
Georgia Tech ,
Organizer: Lei Zhang

In this talk, I will state the main results of center manifold theory for finite dimensional systems and give some simple examples to illustrate their applications. This is based on the book “Applications of Center Manifold Theory” by J. Carr.

Friday, February 26, 2016 - 13:05 ,
Location: Skiles 170 ,
Hongyu Cheng ,
Georgia Tech ,
Organizer: Lei Zhang
We present some basic results from the theory of stochastic processes and investigate the properties of some standard continuous-time stochastic processes. Firstly, we give the deﬁnition of a stochastic process. Secondly, we introduce Brownian motion and study some of its properties. Thirdly, we give some classical examples of stochastic processes in continuous time and at last prove some famous theorems.

Friday, February 19, 2016 - 13:00 ,
Location: Skiles 170 ,
Lei Zhang ,
Georgia Inst. of Technology ,
Organizer: Rafael de la Llave

The Peierls barrier is an observable which characterizes whether the
the set minimizers with a prescribed frequency of a periodic
variational problem form a continuum or have gaps. In solid state
physics Peierls barrier characterizes whether ground states with a
fixed density are pinned or are able to slide. The Peierls barrier
is a microscopic explanation of static friction. Remarkably, in
dynamical systems, Peierls barrier appears also as characterizing
whether KAM circles break down into Cantor sets. Hence, the Peierls
barrier has been investigated both by physicists and by mathematicians
using a variety of methods. We plan to cover the basic definitions of
the variational models and some of the basic results obtainedfrom the
80's. Continuation of last week's seminar

Friday, February 12, 2016 - 13:00 ,
Location: Skiles 170 ,
Lei Zhang ,
Georgia Tech ,
Organizer: Rafael de la Llave

The Peierls barrier is an observable which characterizes whether the the set minimizers with a prescribed frequency of a periodic variational problem form a continuum or have gaps. In solid state physics Peierls barrier characterizes whether ground states with a fixed density are pinned or are able to slide. The Peierls barrier is a microscopic explanation of static friction. Remarkably, in dynamical systems, Peierls barrier appears also as characterizing whether KAM circles break down into Cantor sets. Hence, the Peierls barrier has been investigated both by physicists and by mathematicians using a variety of methods. We plan to cover the basic definitions of the variational models and some of the basic results obtainedfrom the 80's.

Tuesday, November 17, 2015 - 17:00 ,
Location: Skiles 005 ,
Mikel Viana ,
Georgia Tech (Math) ,
Organizer:

In previous talks, we discussed an algorithm (Nash-Moser iteration) to compute invariant whiskered tori for fibered holomorphic maps. Several geometric and number-theoretic conditions are necessary to carry out each step of the iteration. Recently, there has been interest in studying what happens if some of the conditions are removed. In particular, the second Melnikov condition we found can be hard to verify in higher dimensional problems. In this talk, we will use a method due to Eliasson, Moser and Poschel to obtain quasi-periodic solutions which, however, lose an important geometric property relative to the solutions previously constructed.

Tuesday, November 10, 2015 - 17:00 ,
Location: Skiles 005 ,
Rafael de la Llave ,
Georgia Tech ,
Organizer:

In the study of perturbation theories in Dynamical systems one is often interested in solving differential equations involving frequencies satisfying number theoretic properties. We will present some estimates ofsums involving Diophantine frequencies leading to sharp estimates on the differential equations.

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).