Seminars and Colloquia by Series

On numerical composition of Taylor-Fourier

Series
Dynamical Systems Working Seminar
Time
Friday, February 1, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 246
Speaker
Joan Gimeno – BGSMath-UB
A real Taylor-Fourier expression is a Taylor expression whose coefficients are real Fourier series. In this talk we will discuss different numerical methods to compute the composition of two Taylor-Fourier expressions. To this end, we will show some possible implementations and we are going to discuss and show some results in performance. In particular, we are going to cover how the compositon of two Fourier series can be perfomed in logarithmic complexity.

Convergence of the viscosity solutions in vanishing contact structure problem

Series
Dynamical Systems Working Seminar
Time
Friday, January 11, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 246
Speaker
Qinbo Chen – AMSS & GT Math
In this talk, I will discuss the vanishing contact structure problem, which focuses on the asymptotic behavior of the viscosity solutions uε of Hamilton-Jacobi equation H (x, Du(x), ε u(x)) =c, as the factor ε tends to zero. This is a natural generalization of the vanishing discount problem. I will explain how to characterize the limit solution in terms of Peierls barrier functions and Mather measures from a dynamical viewpoint. This is a joint work with Hitoshi Ishii, Wei Cheng, and Kai Zhao.

Scattering maps and instability in Hamiltonian mechanics.

Series
Dynamical Systems Working Seminar
Time
Friday, December 7, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 170
Speaker
Rafael de la Llave – School of Mathematics

Given a Hamiltonian system, normally hyperbolic invariant manifolds and their stable and unstable manifolds are important landmarks that organize the long term behaviour.

When the stable and unstable manifolds of a normally hyperbolic invarriant manifold intersect transversaly, there are homoclinic orbits that converge to the manifold both in the future and in the past. Actually, the orbits are asymptotic both in the future and in the past.

One can construct approximate orbits of the system by chainging several of these homoclinic excursions.

A recent result with M. Gidea and T. M.-Seara shows that if we consider long enough such excursions, there is a true orbit that follows it. This can be considered as an extension of the classical shadowing theorem, that allows to handle some non-hyperbolic directions

A deterministic potential mean-field game

Series
Dynamical Systems Working Seminar
Time
Friday, November 16, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 156
Speaker
Sergio Mayorga – Georgia Tech
In this talk I will begin by discussing the main ideas of mean-field games and then I will introduce one specific model, driven by a smooth hamiltonian with a regularizing potential and no stochastic noise. I will explain what type of solutions can be obtained, and the connection with a notion of Nash equilibrium for a game played by a continuum of players.

A formula with some applications to the theory of Lyapunov exponents (Cancelled)

Series
Dynamical Systems Working Seminar
Time
Friday, November 9, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 156
Speaker
Rui Han – Georgia Tech
We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub, both concerning Lyapunov exponents. This is a work of A. Avila and J. Bochi. https://link.springer.com/article/10.1007/BF02785853

A Simple Analytic Proof for the Shadowing Lemma

Series
Dynamical Systems Working Seminar
Time
Friday, November 2, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 156
Speaker
Yian Yao – GT Math
The Shadowing lemma describes the behaviour of pseudo-orbits near a hyperbolic invariant set. In this talk, I will present an analytic proof of the shadowing lemma for discrete flows. This is a work by K. R. Meyer and George R. Sell.

Invariant Manifolds Associated to Non-resonant Spectral Subspaces II

Series
Dynamical Systems Working Seminar
Time
Friday, October 26, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 156
Speaker
Jiaqi Yang – GT Math
We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace, and some non-resonance conditions are satisfied. Then the map leaves invariant a smooth (as smooth as the map) manifold, which is unique among C^L invariant manifolds. Here, L only depends on the spectrum of the linearization. This is based on a work of Prof. Rafael de la Llave.

Invariant Manifolds Associated to Non-resonant Spectral Subspaces

Series
Dynamical Systems Working Seminar
Time
Friday, October 19, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 156
Speaker
Jiaqi Yang – GT Math
We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace, and some non-resonance conditions are satisfied. Then the map leaves invariant a smooth (as smooth as the map) manifold, which is unique among C^L invariant manifolds. Here, L only depends on the spectrum of the linearization. This is based on a work of Prof. Rafael de la Llave.

A simple proof of a generalization of a Theorem by C.L. Siegel (Part II)

Series
Dynamical Systems Working Seminar
Time
Friday, October 5, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 156
Speaker
Adrian P. Bustamante – Georgia Tech
In this talk I will present a proof of a generalization of a theorem by Siegel, about the existence of an analytic conjugation between an analytic map, $f(z)=\Lambda z +\hat{f}(z)$, and a linear map, $\Lambda z$, in $\mathbb{C}^n$. This proof illustrates a standar technique used to deal with small divisors problems. I will be following the work of E. Zehnder. This is a continuation of last week talk.

Pages