Seminars and Colloquia by Series

Closed Geodesics on Surfaces without Conjugate Points

Series
CDSNS Colloquium
Time
Friday, February 12, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add'l notes for link)
Speaker
Khadim WarIMPA

Please Note: Zoom link: https://zoom.us/j/96065531265?pwd=aW5qZW8vUUt3bGRlN29FS0FFVnc1QT09

We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points. This is based on a join work with Vaughn Climenhaga and Gerhard Knieper.

Forward attractors and limit sets of nonautonomous difference equations

Series
CDSNS Colloquium
Time
Friday, February 5, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add'l notes for link)
Speaker
Peter Kloeden Universität Tübingen

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

The  theory of nonautonomous dynamical systems has undergone  major development during the past 23 years since I talked  about attractors  of nonautonomous difference equations at ICDEA Poznan in 1998. 

Two types of  attractors  consisting of invariant families of  sets   have been defined for  nonautonomous difference equations, one using  pullback convergence with information about the system   in the past and the other using forward convergence with information about the system in the future. In both cases, the component sets are constructed using a pullback argument within a positively invariant  family of sets. The forward attractor so constructed also uses information about the past, which is very restrictive and  not essential for determining future behaviour.  

The forward  asymptotic  behaviour can also be described through the  omega-limit set  of the  system.This set  is closely  related to what Vishik  called the uniform attractor although it need not be invariant. It  is  shown to be asymptotically positively invariant  and also, provided  a future uniformity condition holds, also asymptotically positively invariant.  Hence this omega-limit set provides useful information about  the behaviour in current  time during the approach to the future limit. 

Global solutions for the energy supercritical NLS

Series
CDSNS Colloquium
Time
Friday, January 22, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add'l notes for link)
Speaker
Mouhamadou SyU Virginia

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

In this talk, we will discuss the global well-posedness issue of the defocusing nonlinear Schrödinger equation (NLS). It is known that for subcritical and critical nonlinearities, the equation is globally well-posed on Euclidean spaces and some bounded domains. The supercritical nonlinearities are by far less understood; few partial or conditional results were established. On the other hand, probabilistic approaches (Gibbs measures, fluctuation-dissipation ...) were developed during the last decades to deal with low regularity settings in the context of dispersive PDEs. However, these approaches fail to apply the supercritical nonlinearities.  The aim of this talk is to present a new probabilistic approach recently developed by the author in the context of the energy supercritical NLS. We will review some known results and briefly present earlier probabilistic methods, then discuss the new method and the almost sure global well-posedness consequences for the energy supercritical NLS. The results that will be presented are partly join with Xueying Yu.

 

A von Neumann algebra valued Multiplicative Ergodic Theorem

Series
CDSNS Colloquium
Time
Wednesday, July 22, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Lewis BowenUT Austin

In 1960, Furstenberg and Kesten introduced the problem of describing the asymptotic behavior of products of random matrices as the number of factors tends to infinity. Oseledets’ proved that such products, after normalization, converge almost surely. This theorem has wide-ranging applications to smooth ergodic theory and rigidity theory. It has been generalized to products of random operators on Banach spaces by Ruelle and others. I will explain a new infinite-dimensional generalization based on von Neumann algebra theory which accommodates continuous Lyapunov distribution. No knowledge of von Neumann algebras will be assumed. This is joint work with Ben Hayes (U. Virginia) and Yuqing Frank Lin (UT Austin, Ben-Gurion U.). 

Rapid and Accurate Computation of Invariant Tori, Manifolds, and Connections Near Mean Motion Resonances in Periodically Perturbed Planar Circular Restricted 3-Body Problem Models

Series
CDSNS Colloquium
Time
Wednesday, July 8, 2020 - 12:00 for 1 hour (actually 50 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Bhanu KumarGeorgia Tech

When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, most unstable resonant periodic orbits become invariant tori. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation parameter as well as rotation numbers; 3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds; 4) globalize these manifolds; and 5) compute homoclinic and heteroclinic connections. Our methodology improves on efficiency and accuracy compared to prior studies, and applies to a variety of periodic perturbations. We demonstrate the tools on the planar elliptic RTBP. This is based on joint work with R. Anderson and R. de la Llave.

Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems

Series
CDSNS Colloquium
Time
Wednesday, July 1, 2020 - 09:00 for 1.5 hours (actually 80 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Matteo TanziNew York University

We investigate dynamical systems obtained by coupling  an Anosov diffeomorphism and a N-pole-to-S-pole map of the circle. Both maps are uniformly hyperbolic; however, they have contrasting character, as the first one is chaotic while the second one has “orderly" dynamics. The first thing we show is that even weak coupling can produce interesting phenomena: when the attractor of the uncoupled system is not normally hyperbolic, most small interactions transform it from a smooth surface to a fractal-like set.  We then consider stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. These couplings produce genuine obstructions to uniform hyperbolicity; however, the monotonicity conditions make the system amenable to study by leveraging  techniques from the geometric and ergodic theories of hyperbolic systems.  In particular, we can show existence of invariant cones and SRB measures. 

This is joint work with Lai-Sang Young.

Spectral Galerkin transfer operator methods in uniformly-expanding dynamics

Series
CDSNS Colloquium
Time
Wednesday, June 17, 2020 - 09:00 for 1.5 hours (actually 80 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Caroline WormellUniversity of Sydney

Full-branch uniformly expanding maps and their long-time statistical quantities are commonly used as simple models in the study of chaotic dynamics, as well as being of their own mathematical interest. A wide range of algorithms for computing these quantities exist, but they are typically unspecialised to the high-order differentiability of many maps of interest, and so have a weak tradeoff between computational effort and accuracy.

This talk will cover a rigorous method to calculate statistics of these maps by discretising transfer operators in a Chebyshev polynomial basis. This discretisation is highly efficient: I will show that, for analytic maps, numerical estimates obtained using this discretisation converge exponentially quickly in the order of the discretisation, for a polynomially growing computational cost. In particular, it is possible to produce (non-validated) estimates of most statistical properties accurate to 14 decimal places in a fraction of a second on a personal computer. Applications of the method to the study of intermittent dynamics and the chaotic hypothesis will be presented.

Parameterization of unstable manifolds for delay differential equations

Series
CDSNS Colloquium
Time
Wednesday, June 3, 2020 - 09:00 for 1.5 hours (actually 80 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Jason Mireles-JamesFlorida Atlantic University

Delay differential equations (DDEs) are important in physical applications where there is a time lag in communication between subsystems.  From a mathematical point of view DDEs are an interesting source of problems as they provide natural examples of infinite dimensional dynamical systems.  I'll discuss some spectral numerical methods for computing invariant manifolds for DDEs and present some applications.  

Long-time dynamics for the generalized Korteweg-de Vries and Benjamin-Ono equations

Series
CDSNS Colloquium
Time
Wednesday, May 27, 2020 - 09:00 for 1.5 hours (actually 80 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Benoît GrébertUniversité de Nantes

We provide an accurate description of the long time dynamics for generalized Korteweg-de Vries  and Benjamin-Ono equations on the circle without external parameters and for almost any (in probability and in density) small initial datum. To obtain that result we construct for these two classes of equations and under a very weak hypothesis of non degeneracy of the nonlinearity, rational normal forms on open sets surrounding the origin in high Sobolev regularity. With this new tool we can make precise the long time dynamics of the respective flows. In particular we prove a long-time stability result in Sobolev norm: given a large constant M and a sufficiently small parameter ε, for generic initial datum u(0) of size ε, we control the Sobolev norm of the solution u(t) for time of order ε^{−M}. 

Riemann's non-differentiable function is intermittent

Series
CDSNS Colloquium
Time
Wednesday, May 20, 2020 - 12:00 for 1 hour (actually 50 minutes)
Location
https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Victor da RochaGeorgia Tech

Please Note: (UPDATED Monday 5-18) Note the nonstandard start time of 12PM.

Riemann's non-differentiable function, although introduced as a pathological example in analysis, makes an appearance in a certain limiting regime of the theory of binormal flow for vortex lines. From this physical point of view, it also bears some qualitative similarities to turbulent fluid velocity fields in the infinite Reynolds number limit. In this talk, we'll see how this function arises in the study of the vortex filaments, and how we can adapt the notion of intermittency from the study of turbulent flows to this setting. Then, we'll study the fine intermittent nature of this function on small scales. To do so, we define the flatness, an analytic quantity measuring it, in two different ways. One in the physical space, and the other one in the Fourier space. We prove that both expressions diverge logarithmically as the relevant scale parameter tends to 0, which highlights the (weak) intermittent nature of Riemann's function.

This is a joint work with Alexandre Boritchev (Université de Lyon) and Daniel Eceizabarrena (BCAM, Bilbao).
 

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