Séminaire « Equations différentielles »

In Boltzmann’s billiard a particle moves in a half-plane subject to a central force and is reflected elastically when it hits the boundary. Boltzmann took the central force to be the sum of a gravitational inverse-square-law force and a centrifugal term proportional to the inverse cube of the distance to the centre. He formulated the expectation that except for special values of the parameters the system would be chaotic and would obey his Ergodic Hypothesis.  Recently Gallavotti and Jauslin showed that the system is integrable if the centrifugal term is omitted: it has a second conserved quantity besides the energy. I will review this result and show that this integrable Boltzmann system has the Poncelet property: if in a level set of the conserved quantities a trajectory is periodic then all trajectories on the level set are periodic. As for the classical Poncelet theorem on inscribed-circumscribed polygons in Jacobi’s interpretation, the result relies on the theory of elliptic curves. I will also present some work in progress with Michelle Wang on Boltzmann’s table tennis, the three dimensional version of Boltzmann’s integrable system, and the relation to QRT maps on biquadratic plane curves.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

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Séminaire « Equations différentielles »

The motivic Galois group is most naturally considered as an object in spectral algebraic geometry. However, deep conjectures in the theory of motives imply that the motivic Galois group is classical, i.e., has no higher derived information. We will discuss some recent attempts to verify the classicality of the motivic Galois group.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »
(indiquez vos propres prénom et nom) et laissez le corps du message vide.

A hyperbolic group acts on its Gromov boundary by homeomorphisms. In recent work with Jason Manning, we showed that for groups with sphere boundary, the boundary action is rigid in the sense of topological dynamics: any sufficiently small perturbation is semi-conjugate to the original action. In ongoing work also with Teddy Weisman, we are extending this result to all hyperbolic groups, using a coding argument in the spirit of Sullivan. This talk will introduce the rigidity problem and describe some of the tools towards the proof.

A group endowed with a probability measure comes naturally with a random walk. If moreover this group acts on some space, then one can push the random walk to the space under consideration, up to the choice of a basepoint. The resulting random walk is called the image random walk.

In this talk, motivated by numerous examples (hyperbolic groups acting on one of their Cayley graphs, mapping class groups acting on the corresponding curve complex…), the (discrete) groups will act on Gromov hyperbolic spaces. We will discuss the escape rate for such image random walks, and more precisely the associated large deviations problem.

We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the λφ4 fields over R^4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models’ random current representation, in which the correlation functions’ deviation from Wick’s law is expressed in terms of intersection probabilities of random currents with sources at distances that are large on the model’s lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the λφ4 fields over R^4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models’ random current representation, in which the correlation functions’ deviation from Wick’s law is expressed in terms of intersection probabilities of random currents with sources at distances that are large on the model’s lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the λφ4 fields over R^4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models’ random current representation, in which the correlation functions’ deviation from Wick’s law is expressed in terms of intersection probabilities of random currents with sources at distances that are large on the model’s lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the λφ4 fields over R^4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models’ random current representation, in which the correlation functions’ deviation from Wick’s law is expressed in terms of intersection probabilities of random currents with sources at distances that are large on the model’s lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

Atttention : Les trois premières Leçons auront lieu à l’Institut Mathématique d’Orsay, Amphi Yoccoz, les 1er, 2 et 3 juin

Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :

https://www.fondation-hadamard.fr/fr/financements/accueil-206-cours-avances

Abstract:

Parabolic dynamical systems are mathematical models of the many phenomena which display a « slow » form of chaotic evolution, in the sense that nearby trajectories diverge polynomially in time. 

In contrast with hyperbolic and elliptic dynamical systems, there is no general theory which describes parabolic dynamics. In recent years, a lot of progress has been done in understanding the chaotic features of several classes of such systems, as well as in identifying key mechanisms and techniques which play a central role in their study.

In these lectures we will give a self-contained introduction to some results on chaotic features of parabolic flows, some classical as well as many very recent. We will in particular discuss:

– renormalizable linear flows;

– horocycle flows on compact hyperbolic surfaces and their time-changes;

– the Heisenberg nilflow, nilflows on nilmanifolds and their time-changes;

– smooth area preserving (also known as ‘locally Hamiltonian’) flows on surfaces.

We will define the mathematical objects as well as the dynamical properties we will discuss to keep the lectures accessible to a wide audience and try to highlight throughout the importance of phenomena such as shearing and techniques based on renormalization. Connections between parabolic flows and mathematical physics, spectral theory and Teichmueller dynamics will also be mentioned.

Atttention : Les trois premières Leçons auront lieu à l’Institut Mathématique d’Orsay, Amphi Yoccoz, les 1er, 2 et 3 juin

Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :

https://www.fondation-hadamard.fr/fr/financements/accueil-206-cours-avances

Abstract:

Parabolic dynamical systems are mathematical models of the many phenomena which display a « slow » form of chaotic evolution, in the sense that nearby trajectories diverge polynomially in time. 

In contrast with hyperbolic and elliptic dynamical systems, there is no general theory which describes parabolic dynamics. In recent years, a lot of progress has been done in understanding the chaotic features of several classes of such systems, as well as in identifying key mechanisms and techniques which play a central role in their study.

In these lectures we will give a self-contained introduction to some results on chaotic features of parabolic flows, some classical as well as many very recent. We will in particular discuss:

– renormalizable linear flows;

– horocycle flows on compact hyperbolic surfaces and their time-changes;

– the Heisenberg nilflow, nilflows on nilmanifolds and their time-changes;

– smooth area preserving (also known as ‘locally Hamiltonian’) flows on surfaces.

We will define the mathematical objects as well as the dynamical properties we will discuss to keep the lectures accessible to a wide audience and try to highlight throughout the importance of phenomena such as shearing and techniques based on renormalization. Connections between parabolic flows and mathematical physics, spectral theory and Teichmueller dynamics will also be mentioned.

Atttention : Les trois premières Leçons auront lieu à l’Institut Mathématique d’Orsay, Amphi Yoccoz, les 1er, 2 et 3 juin

Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :

https://www.fondation-hadamard.fr/fr/financements/accueil-206-cours-avances

Abstract:

Parabolic dynamical systems are mathematical models of the many phenomena which display a « slow » form of chaotic evolution, in the sense that nearby trajectories diverge polynomially in time. 

In contrast with hyperbolic and elliptic dynamical systems, there is no general theory which describes parabolic dynamics. In recent years, a lot of progress has been done in understanding the chaotic features of several classes of such systems, as well as in identifying key mechanisms and techniques which play a central role in their study.

In these lectures we will give a self-contained introduction to some results on chaotic features of parabolic flows, some classical as well as many very recent. We will in particular discuss:

– renormalizable linear flows;

– horocycle flows on compact hyperbolic surfaces and their time-changes;

– the Heisenberg nilflow, nilflows on nilmanifolds and their time-changes;

– smooth area preserving (also known as ‘locally Hamiltonian’) flows on surfaces.

We will define the mathematical objects as well as the dynamical properties we will discuss to keep the lectures accessible to a wide audience and try to highlight throughout the importance of phenomena such as shearing and techniques based on renormalization. Connections between parabolic flows and mathematical physics, spectral theory and Teichmueller dynamics will also be mentioned.

Organized by Costas Bachas (LPENS), Semyon Klevtsov (Univ. Strasbourg), Nikita Nekrasov (Simons Center for Geometry and Physics & Stony Brook) and Emmanuel Ullmo (IHES), the « MikeFest: a conference in honor of Michael Douglas » will take place from May 9 to 13, 2022.

Registration until: May 3, 2022.

We are organizing a conference at the IHES on the occasion of Michael R. Douglas’ 60th birthday.

Mike has a long association with IHES, as a visiting professor in 2000-2008 (Louis Michel Chair), by leading the US-based fundraising effort, and as a president and chairman of the Friends of IHES in 2013-2021.

The conference will cover the topics on which Mike has worked and made profound contributions: string theory, matrix models, physical mathematics, and machine learning.

Covid-19 regulations: seats in the conference center are limited to 70 people. The conference will also be available via Zoom.

Invited speakers:

Vijay Balasubramanian, Univ. of Pennsylvania
Alexander Belavin, Independent Univ. Moscow (TBC)
Ilka Brunner, Univ. München
Alain Connes, IHES & Collège de France
Frederik Denef, Columbia Univ.
Bartomeu Fiol, Univ. de Barcelona
Jaume Gomis, Perimeter Institute (TBC)
Sasha Gorsky, IITP RAS, MIPT
Chris Hull, Imperial College
Shamit Kachru, Stanford Univ.
Volodya Kazakov, LPENS
Maxim Kontsevich, IHES
Patrick Massot, LMO – Univ. Paris-Saclay
Luca Mazzucato*, Univ. of Oregon
David McAllester, Toyota Tech. Inst. Chicago (TTIC)*
Liam McAllister, Cornell Univ.
Greg Moore*, Rutgers Univ.
Rémi Monasson, LPENS
Eliezer Rabinovici, Hebrew Univ. of Jerusalem
John Schwarz, CALTECH
Nathan Seiberg*, IAS
Steve Shenker*, Stanford Univ.
Eva Silverstein*, Stanford Univ.
Christian Szegedy, Google Research
Washington Taylor*, MIT
Alessandro Tomasiello, Univ. Degli Studi Di Milano Bicocca
Josef Urban, CIIRC
Steve Zelditch*, Northwestern Univ.

* remote talks