Seminars and Colloquia by Series

Lefschetz Fibrations and Exotic 4-Manifolds

Series
Dynamical Systems Working Seminar
Time
Friday, February 24, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nur Saglam and Jon Simone – Georgia Tech

Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of  exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.

Introduction to the classical Multiplicative Ergodic Theorem

Series
Dynamical Systems Working Seminar
Time
Tuesday, July 21, 2020 - 12:00 for 1 hour (actually 50 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/fsvsfsua
Speaker
Yuqing Lin – UT Austin –

Please Note: This is an expository talk, to be paired with the CDSNS Colloquium held the next day.

 This is a gentle introduction to the classical Oseledets' Multiplicative Ergodic Theorem (MET), which can be viewed as either a dynamical version of the Jordan normal form of a matrix, or a matrix version of the pointwise ergodic theorem (which itself can be viewed as a generalization of the strong law of large numbers).  We will also sketch Raghunathan's proof of the MET and discuss how the MET can be applied to smooth ergodic theory.

Physical Periodic Ehrenfests' Wind-Tree Model

Series
Dynamical Systems Working Seminar
Time
Tuesday, November 19, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hassan Attarchi – GT, School of Math –

We consider a physical periodic Ehrenfests' Wind-Tree model where a moving particle is a hard ball rather than (mathematical) point particle. Some dynamics and statistical properties of this model are studied. Moreover, it is shown that it has a new superdiffusive regime where the diffusion coefficient $D(t)\sim(\ln t)^2$ of dynamics seems to be never observed before in any model.

Existence of a family of solutions in state-dependent delay equations

Series
Dynamical Systems Working Seminar
Time
Tuesday, October 1, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jiaqi Yang – Georgia Tech –
Given an analytic two-dimensional ordinary differential equation which admits a limit cycle, we consider the singular perturbation of it by adding a state-dependent delay. We show that for small enough perturbation, there exist a limit cycle and a two-dimensional family of solutions to the perturbed state-dependent delay equation (SDDE), which resemble the solutions of the original ODE. 
More precisely, for the original ODE, there is a parameterization of the limit cycle and its stable manifold. We show that, there is a very similar parameterization that gives a 2-dimensional family of solutions of the SDDE. 
In our work, we analyze the parameterization, and find functional equations to be satisfied (invariance equations). We prove a theorem in \emph{``a posteriori''} format, that is, if there are approximate solutions of the invariance equations, then there are true solutions of the invariance equations nearby (with appropriate choices of norms). An algorithm which follows from the constructive proof of above theorem has been implemented. 
 
This is a joint work with Joan Gimeno and Rafael de la Llave.

On numerical integrators for state-dependent delay equations

Series
Dynamical Systems Working Seminar
Time
Friday, June 21, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 317
Speaker
Joan Gimeno – Universitat de Barcelona (BGSMath) –
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.

Introduction to KAM theory: II Moser's twist theorem in any dimension

Series
Dynamical Systems Working Seminar
Time
Thursday, May 30, 2019 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Rafael de la Llave – Georigia Inst. of Technology

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. 

Introduction to KAM theory: I the basics.

Series
Dynamical Systems Working Seminar
Time
Wednesday, May 29, 2019 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Rafael de la Llave – Georgia Institute of Technology

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. 

Aubry-Mather theory for homeomorphisms

Series
Dynamical Systems Working Seminar
Time
Friday, April 12, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 246
Speaker
Adrian P. Bustamante – Georgia Tech

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.

spectral equivalence classes based on isospectral reductions

Series
Dynamical Systems Working Seminar
Time
Friday, February 22, 2019 - 03:05 for 1 hour (actually 50 minutes)
Location
Skiles 246
Speaker
Longmei Shu – GT Math
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.

The Proof of an Abstract Nash-Moser Implicit Function Theorem

Series
Dynamical Systems Working Seminar
Time
Friday, February 15, 2019 - 03:05 for 1 hour (actually 50 minutes)
Location
Skiles 246
Speaker
Yian Yao – GT Math
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.

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