# Séminaires : Dynamical systems and PDEs

Géométrie et Dynamique
Sergei Kuksin, Dmitrii Treschev
https://mi-ras-ru.zoom.us/j/98541889798?pwd=SGdnT2lPWCtrbzNjOHQyb09NS0dXdz09 Zoom
Paris 7 Diderot & Steklov Mathematical Institute Online Seminar

## Séances à suivre

+ Séances antérieures

### Séances antérieures

+ Andrey Mironov, Commuting differential and difference operators. 09/06/2021 16:00 https://mi-ras-ru.zoom.us/j/98541889798?pwd=SGdnT2lPWCtrbzNjOHQyb09NS0dXdz09
We will discuss the connection between commuting ordinary differential operators and commuting difference operators. In particular, we construct a discretization of the Lamé operator that preserves the spectral curve.
+ Jacob Bedrossian A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations 26/05/2021 16:00 Zoom'
In a recent joint work with Alex Blumenthal and Sam Punshon-Smith, we put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a degenerate Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a quantitative version of Hörmander’s hypoelliptic regularity theory in an L1 framework which estimates this Fisher information from below by a fractional Sobolev norm using the Kolmogorov equation.. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE and that this class includes the Lorenz 96 model in any dimension greater than or equal to 7 (as well as finite-dimensional truncations of shell models GOY and SABRA). This is the first mathematically rigorous proof of chaos (in the sense of positive Lyapunov exponents) for stochastically driven Lorenz 96, despite the overwhelming numerical evidence (the deterministic case remains far out of reach).
+ Dario Bambusi Growth of Sobolev norms for unbounded perturbations of the Laplacian on flat tori (towards a quantum Nekhoroshev theorem) 12/05/2021 16:00 https://mi-ras-ru.zoom.us/j/98541889798?pwd=SGdnT2lPWCtrbzNjOHQyb09NS0dXdz09
I will present a study of the time dependent Schroedinger equation $$-i \dot\psi =-\Delta\psi +F(tex,-i\nabla)\psi,$$ on a flat $d$ dimensional torus. Here $F$ is a time dependent pseudodifferential operator of order strictly smaller than 2. The main result I will give is an estimate ensuring that the Sobolev norms of the solutions are bounded by $t^\epsilon$. The proof is a quantization of the proof of the Nekhoroshev theorem, both analytic and geometric parts. Previous results of this kind were limited either to the case of bounded perturbations of the Laplacian or to quantization of systems with a trivial geometry of the resonances, like harmonic oscillators or 1-d systems. In this seminar I will present the result and the main ideas of the proof.
+ François Huveneer Integrability breaking in extended Hamiltonian systems 21/04/2021 16:00 https://mi-ras-ru.zoom.us/j/98541889798?pwd=SGdnT2lPWCtrbzNjOHQyb09NS0dXdz09
Format: This time the talk will consist of two parts 50 minutes each, with a short break in between. The first part is supposed to be more introductory than the second. Abstract: In low dimensional Hamiltonian systems, several classical results such as the KAM theorem or Nekhoroshev estimates guarantee that the dynamics remains close to integrable in the vicinity of an integrable point. In statistical physics and thermodynamics, one needs to consider extensive systems at positive temperature. In this case, the common belief is that integrability is completely lost as soon as one leaves the integrable limit. Nevertheless, as we will see in this talk, the dynamics on some intermediate time scales may be strongly affected by integrable effects. The understanding of the dynamics on such timescales is directly relevant to evaluate the thermal or electrical conductivity. The talk will consist of two parts: First, I will introduce the topic, describe the phenomenology and state a few mathematical results that we obtained in the last years (works with W. De Roeck). Second, I will discuss the Green-Kubo formula for the conductivity and I will present some open problems related to our results. Link: https://mi-ras-ru.zoom.us/j/98541889798?pwd=SGdnT2lPWCtrbzNjOHQyb09NS0dXdz09 Password: 933727
+ On the long-term dynamics of nonlinear wave equations and the uniqueness of solitons 07/04/2021 16:00
+ Alexander VESELOV Geodesic scattering on hyperboloids and Knoerrer’s map 24/03/2021 16:00
Geodesic flow on ellipsoids is one of the most celebrated classical integrable systems considered by Jacobi in 1837.Moser revisited this problem in 1978 revealing the link with the modern theory of solitons.Surprisingly a similar question for hyperboloids did not get much attention, although the dynamics in this case is very different. I will explain how to use the remarkable results of Moser and Knoerrer on the relations between Jacobi problem and integrable Neumann system on sphere to describe explicitly the geodesic scattering on hyperboloids. It will be shown also that Knoerrer's reparametrisation is closely related to the projectively equivalent metric on a quadric discovered in 1998 by Tabachnikov and, independently, by Matveev and Topalov, giving a new proof of their result. The projectively equivalent metric (in contrast to the usual one) turns out to be regular on the projective closure of hyperboloid, which allows us to extend Knoerrer's map to this closure. The talk is based on a recent joint work with Lihua Wu.
+ Laurent Stolovitch Geometry of hyperbolic Cauchy-Riemann singularities and KAM-like theory for holomorphic involutions 10/03/2021 16:00
In this talk, we emphasize how the understanding of the properties of some dynamical systems can lead to the understandings of some (a priori unrelated) geometric problems. To be more specific, we shall give some new insights of the geometry of germs of real analytic surfaces in (C^2,0) having an isolated Cauchy-Riemann (CR) singularity at the origin. These are perturbations of Bishop quadrics. There are two kinds of CR singularities stable under perturbations: elliptic and hyperbolic. Elliptic case was studied by Moser-Webster in their seminal '83 article, where they showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric/analytic features can be read off.Here, we focus on perturbations of hyperbolic quadrics. As was shown by Moser-Webster, such a surface can be transformed to a formal normal form by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a non-degenerate real analytic surface M in (C^2,0) having a hyperbolic CR singularity at the origin, we prove the existence of Whitney smooth family of holomorphic curves intersecting M along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions $\{\tau_1,\tau_2\}$ at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to $\{z_1z_2=const\}$ (which is not the usual torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets. This is a joint work with Z. Zhao.
+ Victor Kleptsyn The Furstenberg theorem: adding a parameter and removing the stationarity (On a joint work with A. Gorodetski) 24/02/2021 16:00
The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices from SL(n,R); their norms turn out to grow exponentially. In our joint work, we study what happens if the random matrices from SL(2,R) depend on an additional parameter. It turns out that in this new situation, the conclusion changes. Namely, under some natural conditions, there almost surely exists a (random) "exceptional" set of parameters where the lower limit for the Lyapunov exponent vanishes. Another direction of the generalization of the classical Furstenberg theorem is removing the stationarity assumption. That is, the matrices that are multiplied are still independent, but no longer identically distributed. Though in this setting most of the standard tools are no longer applicable (no more stationary measure, no more Birkhoff ergodic theorem, etc.), it turns out that the Furstenberg theorem can (under the appropriate assumptions) still be generalized to this setting, with a deterministic sequence replacing the Lyapunov exponent. These two generalizations can be mixed together, providing the Anderson localization conclusions for the non-stationary 1D random Schrodinger operators.
+ Roman Novikov Multidimensional inverse scattering problem for the Schrödinger equation 10/02/2021 16:00
We give a short review of old and recent results on the multidimensional inverse scattering problem for the Schrödinger equation. A special attention is paid to efficient reconstructions of the potential from scattering data which can be measured in practice. Potential applications include phaseless inverse X-ray scattering, acoustic tomography and tomographies using elementary particles.
+ Isabelle Gallagher On the derivation of the Boltzmann equation: fluctuations and large deviations 27/01/2021 16:00
It has been known since Lanford’s result in 1974 that in the limit when the number of particles goes to infinity in a rarefied gas, the one-particle distribution satisfies the Boltzmann equation, at least for a short time. In this talk we shall analyze the fluctuations, and large deviations around that limit. This corresponds to joint works with Thierry Bodineau, Laure Saint-Raymond and Sergio Simonella.
+ Dmitry Treschev Entropy of an operator 13/01/2021 16:00
We extend the concept of the measure entropy from the group of automorphisms of a measure space to the group of unitary operators on a Hilbert space. Our main motivations concern formalization of the idea of quantum chaos. The key ingredient of our construction is a (probably) new concept from functional analysis, the mu-norm of a (bounded) operator on a Hilbert space.
+ Leonid Polterovich "Instabilities in Hamiltonian dynamics and symplectic topology" 23/12/2020 16:00
I outline an existence mechanism, based on symplectic topology, for orbits of Hamiltonian flows connecting a pair of disjoint subsets in the phase space. Applications include "superconductivity channels" in nearly integrable systems, and contact dynamics (joint with Michael Entov).
+ Laurent Niederman (IMCCE-Orsay) Quasi periodic co-orbital motions 09/12/2020 16:00
This is a joint work with Philippe Robutel and Alexandre Pouss
+ Sergey Denisov (University of Wisconsin-Madison) , Singularity formation in the contour dynamics for 2d Euler equation on the plane 25/11/2020 16:00
Abstract: We will study 2d Euler dynamics of centrally symmetric pair of patches on the plane. In the presence of exterior regular velocity, we will show that these patches can merge so fast that the distance between them allows double-exponential upper bound which is known to be sharp. The formation of the 90 degree corners on the interface and the self-similarity analysis of this process will be discussed. For a model equation, we will discuss existence of the curve of smooth stationary solutions that originates at singular stationary steady state.
+ S. L. Tabachnikov Flavors of bicycle mathematics 28/10/2020 16:00
This talk concerns a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment. Surprisingly, this simple model is quite rich and has connections with several areas of research, including completely integrable systems. Here is a sampler of problems that I hope to touch upon: 1) The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences. 2) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. These two front tracks are related by the bicycle (Backlund, Darboux) correspondence, which defines a discrete time dynamical system on the space of curves. This system is completely integrable and it is closely related with another, well studied, completely integrable dynamical system, the filament (a.k.a binormal, smoke ring, local induction) equation. 3) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is an open problem, intimately related with Ulam's problem in flotation theory (in dimension two): is the round ball the only body that floats in equilibrium in all positions? This problem is also related to the motion of a charge in a magnetic field of a special kind. It turns out that the known solutions are solitons of the planar version of the filament equation.