Seminars and Colloquia by Series

Introduction to stochastic processes

Series
Dynamical Systems Working Seminar
Time
Friday, February 26, 2016 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 170
Speaker
Hongyu Cheng – Georgia Tech
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.

The Peierls barrier in one-dimensional models II

Series
Dynamical Systems Working Seminar
Time
Friday, February 19, 2016 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 170
Speaker
Lei Zhang – Georgia Inst. of Technology
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

The Peierls barrier in one dimensional models

Series
Dynamical Systems Working Seminar
Time
Friday, February 12, 2016 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 170
Speaker
Lei Zhang – Georgia Tech
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.

Almost-reducibility for fibered holomorphic dynamics

Series
Dynamical Systems Working Seminar
Time
Tuesday, November 17, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mikel Viana – Georgia Tech (Math)
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.

Sums involving Diophantine numbers and applications to differential equations.

Series
Dynamical Systems Working Seminar
Time
Tuesday, November 10, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rafael de la Llave – Georgia Tech
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.

Construction of whiskered invariant tori for fibered holomorphic dynamics II

Series
Dynamical Systems Working Seminar
Time
Thursday, October 29, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mikel Viana – Georgia Tech (Math)
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.

Construction of whiskered invariant tori for fibered holomorphic dynamics (I: Reducibility).

Series
Dynamical Systems Working Seminar
Time
Tuesday, October 6, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mikel de Viana – Georgia Tech
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.

Whitney differentiability in KAM theory II

Series
Dynamical Systems Working Seminar
Time
Tuesday, September 29, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rafael de la Llave – Georgia Tech (Math)
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).

Whitney differentiability in KAM theory

Series
Dynamical Systems Working Seminar
Time
Tuesday, September 22, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Rafael de la Llave – Georgia Institute of Technology
We will review the notion of Whitney differentiability and the Whitney embedding theorem. Then, we will also review its applications in KAM theory.

Construction of quasi-periodic solutions of State-dependent delay differential equations by the parameterization method II: Details.

Series
Dynamical Systems Working Seminar
Time
Tuesday, September 15, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xiaolong He – Georgia Tech (Math)/Hunan University
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.

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