Seminars and Colloquia by Series

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.

Langevin dynamics with manifold structure: efficient solvers and predictions for conformational transitions in high dimensions

Series
Applied and Computational Mathematics Seminar
Time
Monday, June 22, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/963540401
Speaker
Dr. Yuan GaoDuke University

Please Note: virtual (online) seminar

We work on Langevin dynamics with collected dataset that distributed on a manifold M in a high dimensional Euclidean space. Through the diffusion map, we learn the reaction coordinates for N which is a manifold isometrically embedded into a low dimensional Euclidean space. This enables us to efficiently approximate the dynamics described by a Fokker-Planck equation on the manifold N. Based on this, we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of N and enjoys the weighted l^2 convergence to the Fokker-Planck equation. The proposed upwind scheme leads to a Markov chain with transition probability between the nearest neighbor points, which enables us to directly conduct manifold-related computations such as finding the optimal coarse-grained network and the minimal energy path that represents chemical reactions or conformational changes. To acquire information about the equilibrium potential on manifold N, we apply a Gaussian Process regression algorithm to generate equilibrium potentials for a new physical system with new parameters. Combining with the proposed upwind scheme, we can calculate the trajectory of the Fokker-Planck equation on N based on the generated equilibrium potential. Finally, we develop an algorithm to pullback the trajectory to the original high dimensional space as a generative data for the new physical system. This is a joint work with Nan Wu and Jian-Guo Liu.

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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