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Friday, June 21, 2019 - 14:00 ,
Location: Skiles 317 ,
Joan Gimeno ,
Universitat de Barcelona (BGSMath) ,
joan@maia.ub.es ,
Organizer: Yian Yao

Abstract: Many real-life phenomena in science can be modeled by an Initial Value Problem (IVP) for ODE's. To make the model more consistent with real phenomenon, it sometimes needs to include the dependence on past values of the state. Such models are given by retarded functional differential equations. When the past values depend on the state, the IVP is not always defined. Several examples illustrating the problems and methods to integrate IVP of these kind of differential equations are going to be explained in this talk.

Thursday, May 30, 2019 - 14:00 ,
Location: Skiles 005 ,
Rafael de la Llave ,
Georigia Inst. of Technology ,
Organizer:

he KAM (Kolmogorov Arnold and Moser) theory studies

the persistence of quasi-periodic solutions under perturbations.

It started from a basic set of theorems and it has grown

into a systematic theory that settles many questions.

The basic theorem is rather surprising since it involves delicate

regularity properties of the functions considered, rather

subtle number theoretic properties of the frequency as well

as geometric properties of the dynamical systems considered.

In these lectures, we plan to cover a complete proof of

a particularly representative theorem in KAM theory.

In the first lecture we will cover all the prerequisites

(analysis, number theory and geometry). In the second lecture

we will present a complete proof of Moser's twist map theorem

(indeed a generalization to more dimensions).

The proof also lends itself to very efficient numerical algorithms.

If there is interest and energy, we will devote a third lecture

to numerical implementations.

Wednesday, May 29, 2019 - 14:00 ,
Location: Skiles 005 ,
Rafael de la Llave ,
Georgia Institute of Technology ,
Organizer: Yian Yao

The KAM (Kolmogorov Arnold and Moser) theory studies

the persistence of quasi-periodic solutions under perturbations.

It started from a basic set of theorems and it has grown

into a systematic theory that settles many questions.

The basic theorem is rather surprising since it involves delicate

regularity properties of the functions considered, rather

subtle number theoretic properties of the frequency as well

as geometric properties of the dynamical systems considered.

In these lectures, we plan to cover a complete proof of

a particularly representative theorem in KAM theory.

In the first lecture we will cover all the prerequisites

(analysis, number theory and geometry). In the second lecture

we will present a complete proof of Moser's twist map theorem

(indeed a generalization to more dimensions).

The proof also lends itself to very efficient numerical algorithms.

If there is interest and energy, we will devote a third lecture

to numerical implementations.

Friday, April 12, 2019 - 15:05 ,
Location: Skiles 246 ,
Adrian P. Bustamante ,
Georgia Tech ,
Organizer: Adrian Perez Bustamante

In this talk we will follow the paper titled "Aubry-Mather theory for homeomorphisms", in which it is developed a variational approach to study the dynamics of a homeomorphism on a compact metric space. In particular, they are described orbits along which any Lipschitz Lyapunov function has to be constant via a non-negative Lipschitz semidistance. This is work of Albert Fathi and Pierre Pageault.

Friday, February 22, 2019 - 03:05 ,
Location: Skiles 246 ,
Longmei Shu ,
GT Math ,
Organizer: Jiaqi Yang

Isospectral reductions on graphs remove certain nodes and change the weights of remaining edges. They preserve the eigenvalues of the adjacency matrix of the graph, their algebraic multiplicities and geometric multiplicities. They also preserve the eigenvectors. We call the graphs that can be isospectrally reduced to one same graph spectrally equivalent. I will give examples to show that two graphs can be spectrally equivalent or not based on the feature one picks for the equivalence class.

Friday, February 15, 2019 - 03:05 ,
Location: Skiles 246 ,
Yian Yao ,
GT Math ,
Organizer: Jiaqi Yang

I will present a proof of an abstract Nash-Moser Implicit Function Theorem. This theorem can cope with derivatives which are not boundly invertible from one space to itself. The main technique is to combine Newton steps - which loses derivatives with some smoothing that restores them.

Friday, February 1, 2019 - 15:05 ,
Location: Skiles 246 ,
Joan Gimeno ,
BGSMath-UB ,
Organizer: Jiaqi Yang

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.

Friday, January 11, 2019 - 15:00 ,
Location: Skiles 246 ,
Qinbo Chen ,
AMSS &amp; GT Math ,
Organizer:

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.

Friday, December 7, 2018 - 15:00 ,
Location: Skiles 170 ,
Rafael de la Llave ,
School of Mathematics ,
Organizer: Rafael de la Llave

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

Friday, November 16, 2018 - 15:05 ,
Location: Skiles 156 ,
Sergio Mayorga ,
Georgia Tech ,
Organizer: Jiaqi Yang

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.