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

I will describe the proof of the crossing property for the 4-point function.

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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 indoor
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 70 persons in 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_physique PRENOM NOM »
(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Séminaire « Equations différentielles »

The Mahler measure of multivariate polynomials appears in several fields of mathematics including number theory. In particular there are deep conjectural links between Mahler measures of integer polynomials and special values of L-functions. The first theorem in this direction is due to Smyth (1981), who computed the Mahler measure of the polynomial 1+x+y in terms of the L-function of the Dirichlet character of conductor 3. This can be generalised to a class of 2-variable polynomials called exact polynomials. I will explain work in progress with Riccardo Pengo, where we introduce the degree of exactness of a polynomial P(x_1,…,x_n). Under certain conditions, the Mahler measure of P can then be interpreted as a Deligne period on a subvariety of the hypersurface P=0, whose codimension is equal to the exactness degree of P.

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

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 70 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 70 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.

In 2006, Chapoton defined a class of Tamari intervals called « new intervals » in his enumeration of Tamari intervals, and he found that these new intervals are equi-enumerated with bipartite planar maps. We present here a direct bijection between these two classes of objects using a new object called « degree tree ». Our bijection also gives an intuitive proof of an unpublished equi-distribution result of some statistics on new intervals given by Chapoton and Fusy.

A long-standing problem proposed by David Aldous consists in giving a sharp upper bound for the mixing time of the so-called “triangulation walk”, a Markov chain defined on the set of all possible triangulations of the regular n-gon. A single step of the chain consists in performing a random edge flip, i.e. in choosing an (internal) edge of the triangulation uniformly at random and, with probability 1/2, replacing it with the other diagonal of the quadrilateral formed by the two triangles adjacent to the edge in question (with probability 1/2, the triangulation is left unchanged).
While it has been shown that the relaxation time for the triangulation walk is polynomial in n and bounded below by a multiple of n^{3/2}, the conjectured sharpness of the lower bound remains firmly out of reach in spite of the apparent simplicity of the chain. For edge flip chains on different models — such as planar maps, quadrangulations of the sphere, lattice triangulations and other geometric graphs — even less is known.
We shall discuss results concerning the mixing time of random edge flips on rooted quadrangulations of the sphere, partly obtained in joint work with Alexandre Stauffer. A “growth scheme” for quadrangulations which generates a uniform quadrangulation of the sphere by adding faces one at a time at appropriate random locations can be combined with careful combinatorial constructions to build probabilistic canonical paths in a relatively novel way. This method has immediate implications for a range of interesting edge-manipulating Markov chains on so-called Catalan structures, from “leaf translations” on plane trees to “edge rotations” on general planar maps. Moreover, we are able to apply it to flips on 2p-angulations and simple triangulation of the sphere, via newly developed “growth schemes”.