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Post-edited  Interview au Cirm : Fabien Durand
Durand, Fabien (Personne interviewée) | CIRM (Editeur )

Interview de Fabien Durand, mathématicien à l'Université de Picardie Jules Verne, président de la Société Mathématique de France depuis le 1er juillet 2020.

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Post-edited  Interview at Cirm : Robert Tichy
Tichy, Robert (Personne interviewée) | CIRM (Editeur )

Robert Franz Tichy is an Austrian mathematician and professor at Graz University of Technology.

He studied mathematics at the University of Vienna and finished with a Ph.D. thesis on uniform distribution under the supervision of Edmund Hlawka. He received his habilitation at TU Wien in 1983. Currently he is a professor at the Institute for Analysis and Number Theory at TU Graz. Previous positions include head of the Department of Mathematics and Dean of the Faculty of Mathematics, Physics and Geodesy at TU Graz, President of the Austrian Mathematical Society, and Member of the Board (Kuratorium) of the FWF, the Austrian Science Foundation.

His research deals with Number theory, Analysis and Actuarial mathematics, and in particular with number theoretic algorithms, digital expansions, diophantine problems, combinatorial and asymptotic analysis, quasi Monte Carlo methods and actuarial risk models. Among his contributions are results in discrepancy theory, a criterion (joint with Yuri Bilu) for the finiteness of the solution set of a separable diophantine equation, as well as investigations of graph theoretic indices and of combinatorial algorithms with analytic methods. He also investigated (with Istvan Berkes and Walter Philipp) pseudorandom properties of lacunary sequences.
In 1985 he received the Prize of the Austrian Mathematical Society. Since 2004 he has been a Corresponding Member of the Austrian Academy of Sciences. In 2017 he received an honorary doctorate from the University of Debrecen. He taught as a visiting professor at the University of Illinois at Urbana-Champaign and the Tata Institute of Fundamental Research. In 2017 he was a guest professor at Paris 7; currently (until February 2021) he holds the Morlet chair at the Centre International de Rencontres Mathématiques in Luminy (https://www.chairejeanmorlet.com/2020...​)
Robert Franz Tichy is an Austrian mathematician and professor at Graz University of Technology.

He studied mathematics at the University of Vienna and finished with a Ph.D. thesis on uniform distribution under the supervision of Edmund Hlawka. He received his habilitation at TU Wien in 1983. Currently he is a professor at the Institute for Analysis and Number Theory at TU Graz. Previous positions include head of the Department of Mathematics ...

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Andreï Kolmogorov est un mathématicien russe (1903-1987) qui a apporté des contributions frappantes en théorie des probabilités, théorie ergodique, turbulence, mécanique classique, logique mathématique, topologie, théorie algorithmique de l'information et en analyse de la complexité des algorithmes. Alexander Bufetov, Directeur de recherche CNRS (I2M - Aix-Marseille Université, CNRS, Centrale Marseille) et porteur local de la Chaire Jean-Morlet (Chaire Tamara Grava 2019 - semestre 1) donnera une conférence sur les contributions exceptionnelles et la vie dramatique d'un grand génie du XXe siècle.
Andreï Kolmogorov est un mathématicien russe (1903-1987) qui a apporté des contributions frappantes en théorie des probabilités, théorie ergodique, turbulence, mécanique classique, logique mathématique, topologie, théorie algorithmique de l'information et en analyse de la complexité des algorithmes. Alexander Bufetov, Directeur de recherche CNRS (I2M - Aix-Marseille Université, CNRS, Centrale Marseille) et porteur local de la Chaire Jean-Morlet ...

00A06 ; 00A09 ; 01Axx ; 01A60

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Multi angle  Interview au Cirm : Fabien Durand & Samuel Petite
Durand, Fabien (Personne interviewée) ; Petite, Samuel (Personne interviewée) | CIRM (Editeur )

This conference will gather researchers working on different topics such as combinatorics, computer science, probability, geometry, physics, quasicrystallography, ... but sharing a common interest: dynamical systems and more precisely subshifts, tilings and group actions. It will focus on algebraic and dynamical invariants such as group automorphisms, growth of symbolic complexity, Rauzy graphs, dimension groups, cohomology groups, full groups, dynamical spectrum, amenability, proximal pairs, ... With this conference we aim to spread out these invariants outside of their original domains and to deepen their connections with combinatorial and dynamical properties.
This conference will gather researchers working on different topics such as combinatorics, computer science, probability, geometry, physics, quasicrystallography, ... but sharing a common interest: dynamical systems and more precisely subshifts, tilings and group actions. It will focus on algebraic and dynamical invariants such as group automorphisms, growth of symbolic complexity, Rauzy graphs, dimension groups, cohomology groups, full groups, ...

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Pants complexes of large surfaces were proved to be vigid by Margalit. We will consider convergence completions of curve and pants complexes and show that some weak four of rigidity holds for the latter. Some key tools come from the geometry of Deligne Mumford compactification of moduli spaces of curves with level structures.

57M10 ; 20F34 ; 14D23

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Multi angle  Local systems and Satake duality
Fock, Vladimir (Auteur de la Conférence) | CIRM (Editeur )

Satake duality is an isomorphism between the algebra of characters of a simple Lie group G and the Hecke algebra of the affine Lie group corresponding to the Langlands dual to G. We will suggest a (conjectural) generalisation of this isomorphism replacing the characters by functions on local systems on surfaces and Hecke algebra by the algebra on cells on local systems with values in an affine Lie group.

17B20 ; 30F60

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Multi angle  Spectra of ultra-discrete limits
Zuk, Andrzej (Auteur de la Conférence) | CIRM (Editeur )

We present a computation of spectra of random walks on self-similar graphs.

37A30 ; 05C25 ; 35Q53 ; 20M35

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Many effects of climate change seem to be reflected not in the mean temperatures, precipitation or other environmental variables, but rather in the frequency and severity of the extreme events in the distributional tails. The most serious climate-related disasters are caused by compound events that result from an unfortunate combination of several variables. Detecting changes in size or frequency of such compound events requires a statistical methodology that efficiently uses the largest observations in the sample.
We propose a simple, non-parametric test that decides whether two multivariate distributions exhibit the same tail behavior. The test is based on the entropy, namely Kullback-Leibler divergence, between exceedances over a high threshold of the two multivariate random vectors. We study the properties of the test and further explore its effectiveness for finite sample sizes.
Our main application is the analysis of daily heavy rainfall times series in France (1976 -2015). Our goal in this application is to detect if multivariate extremal dependence structure in heavy rainfall change according to seasons and regions.
Many effects of climate change seem to be reflected not in the mean temperatures, precipitation or other environmental variables, but rather in the frequency and severity of the extreme events in the distributional tails. The most serious climate-related disasters are caused by compound events that result from an unfortunate combination of several variables. Detecting changes in size or frequency of such compound events requires a statistical ...

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It is generally admitted that financial time series have heavy tailed marginal distributions. When time series models are fitted on such data, the non-existence of appropriate moments may invalidate standard statistical tools used for inference. Moreover, the existence of moments can be crucial for risk management. This talk considers testing the existence of moments in the framework of standard and augmented GARCH models. In the case of standard GARCH, even-moment conditions involve moments of the independent innovation process. We propose tests for the existence of moments of the returns process that are based on the joint asymptotic distribution of the estimator of the volatility parameters and empirical moments of the residuals. To achieve efficiency gains we consider non Gaussian QML estimators founded on reparametrizations of the GARCH model, and we discuss optimality issues. We also consider augmented GARCH processes, for which moment conditions are less explicit. We establish the asymptotic distribution of the empirical moment Generating function (MGF) of the model, defined as the MGF of the random autoregressive coefficient in the volatility dynamics, from which a test is deduced. An alternative test is based on the estimation of the maximal exponent characterizing the existence of moments. Our results will be illustrated with Monte Carlo experiments and real financial data.
It is generally admitted that financial time series have heavy tailed marginal distributions. When time series models are fitted on such data, the non-existence of appropriate moments may invalidate standard statistical tools used for inference. Moreover, the existence of moments can be crucial for risk management. This talk considers testing the existence of moments in the framework of standard and augmented GARCH models. In the case of ...

37M10 ; 62M10 ; 62P20

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I will describe a program to describe Hitchin components as the moduli space of some new geometric structure on the surface. This geometric structure generalizes the complex structure. Its construction uses the punctual Hilbert scheme of the plane. It should give a unified description of Hitchin components without fixed complex structure on the surface. I also present a generalization to character varieties for non split real groups in the spirit of G-Higgs bundles.
I will describe a program to describe Hitchin components as the moduli space of some new geometric structure on the surface. This geometric structure generalizes the complex structure. Its construction uses the punctual Hilbert scheme of the plane. It should give a unified description of Hitchin components without fixed complex structure on the surface. I also present a generalization to character varieties for non split real groups in the ...

30F60 ; 14D21 ; 53C15 ; 14C05

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Multi angle  Equivalent curves on surfaces
Xu, Binbin (Auteur de la Conférence) | CIRM (Editeur )

We consider a closed oriented surface of genus at least 2. For any positive integer k, an essential closed curve on the surface with k self-intersections is called a k-curve. A pair of curves on the surface are said to be k-equivalent, if they have the same intersection numbers with each k-curve. In this talk, I will discuss the general picture of a pair of k-equivalent curves and the relation between k-equivalence relations for different k’s.
This is a joint-work with Hugo Parlier
We consider a closed oriented surface of genus at least 2. For any positive integer k, an essential closed curve on the surface with k self-intersections is called a k-curve. A pair of curves on the surface are said to be k-equivalent, if they have the same intersection numbers with each k-curve. In this talk, I will discuss the general picture of a pair of k-equivalent curves and the relation between k-equivalence relations for different k’s. ...

57M99

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I will present a recent result in the theory of unitary representations of lattices in semi-simple Lie groups, which can be viewed as simultaneous generalization of Margulis normal subgroup theorem and C*-simplicity and the unique trace property for such lattices. The strategy of proof gathers ideas of both of these results: we extend Margulis’ dynamical approach to the non-commutative setting, and apply this to the conjugation dynamical system induced by a unitary representation. On the way, we obtain a new proof of Peterson’s character rigidity result, and a new rigidity result for uniformly recurrent subgroups of such lattices. I will give some basics on non-commutative ergodic theory and explain-some steps to prove the main result and its applications. This is based on joint works with Uri Bader, Cyril Houdayer, and Jesse Peterson.
I will present a recent result in the theory of unitary representations of lattices in semi-simple Lie groups, which can be viewed as simultaneous generalization of Margulis normal subgroup theorem and C*-simplicity and the unique trace property for such lattices. The strategy of proof gathers ideas of both of these results: we extend Margulis’ dynamical approach to the non-commutative setting, and apply this to the conjugation dynamical system ...

22D10 ; 22D25 ; 22E40 ; 46L10 ; 46L30

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The study of group actions on Hilbert spaces is central in operator algebras, geometric group theory and representation theory. In many natural situations however, particularily interesting actions on Lp spaces appear for p not 2. One celebrated example is the construction by Pansu (and later greatly generalized by Yu to all Gromov hyperbolic groups) of proper actions of groups of isometries of hyperbolic spaces on Lp for large p. In all these results, the rather clear impression was that it was easier to act on Lp space as p becomes larger. The goal of my talk will be to explain this impression by a theorem and to study how the behaviour of the group actions on Lp spaces depends on p and on the group. In particular, I will show that the set of values of p such that a given countable groups has an isometric action on Lp with unbounded orbits is of the form $[p_c,\infty]$ for some $p_c$, and I will try to compute this critical parameter for lattices in semisimple groups. In passing, we will have to discuss how these objects and properties behave with respect to quantitative measure equivalence. This is a joint work with Amine Marrakchi, partly in arXiv:2001.02490.
The study of group actions on Hilbert spaces is central in operator algebras, geometric group theory and representation theory. In many natural situations however, particularily interesting actions on Lp spaces appear for p not 2. One celebrated example is the construction by Pansu (and later greatly generalized by Yu to all Gromov hyperbolic groups) of proper actions of groups of isometries of hyperbolic spaces on Lp for large p. In all these ...

22F05 ; 46C05

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Given a finite simple graph X, the right-angled Artin group associated to X is defined by the following very simple presentation: it has one generator per vertex of X, and the only relations consist in imposing that two generators corresponding to adjacent vertices commute. We investigate right-angled Artin groups from the point of view of measured group theory. Our main theorem is that two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. On the other hand, right-angled Artin groups are never superrigid from this point of view: given any right-angled Artin group G, I will also describe two ways of producing groups that are measure equivalent to G but not commensurable to G.This is joint work with Jingyin Huang.
Given a finite simple graph X, the right-angled Artin group associated to X is defined by the following very simple presentation: it has one generator per vertex of X, and the only relations consist in imposing that two generators corresponding to adjacent vertices commute. We investigate right-angled Artin groups from the point of view of measured group theory. Our main theorem is that two right-angled Artin groups with finite outer au...

20F36 ; 20F65 ; 37A20

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The Gaussian functor associates to every orthogonal representation of a group G on a Hilbert space, a probability measure preserving action of G called a Gaussian action. This construction is a fundamental tool in ergodic theory and is the source of a large and interesting class of probability measure preserving actions. In this talk, I will present a generalization of the Gaussian functor which associates to every affine isometric action of G on a Hilbert space, a nonsingular Gaussian action which is not measure preserving. This provides a new and large class of nonsingular actions whose properties are related in a very subtle way to the geometry of the original affine isometric action. In some cases, such as affine isometric actions comming from groups acting on trees, a fascinating phase transition phenomenon occurs.This talk is based on a joint work with Yuki Arano and Yusuke Isono, as well as a more recent joint work with Stefaan Vaes.
The Gaussian functor associates to every orthogonal representation of a group G on a Hilbert space, a probability measure preserving action of G called a Gaussian action. This construction is a fundamental tool in ergodic theory and is the source of a large and interesting class of probability measure preserving actions. In this talk, I will present a generalization of the Gaussian functor which associates to every affine isometric action of G ...

37A40 ; 20E08 ; 20F65 ; 28C20 ; 37A50

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We prove uniqueness of the solutions ($u$, velocity and $\theta$, temperature) of the Boussinesq system in the whole space ${\mathbb{R}}^3$ in the critical functional spaces: continuous in time with values in $L^3$ for the velocity and $L^2$ in time with values in $L^{3/2}$ in space for the temperature. The proof relies on the property of maximal regularity for the heat equation.

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The purpose of this talk is to present two 1d congestion models: a soft congestion model with a singular pressure, and a hard congestion model in which the dynamic is different in the congested and non-congested zone (incompressible vs. compressible dynamic). The hard congested model is the limit of the soft one as the parameter within the singular presure vanishes.
For each model, we prove the existence of traveling waves, and we study their stability. This is a joint work with Charlotte Perrin.
The purpose of this talk is to present two 1d congestion models: a soft congestion model with a singular pressure, and a hard congestion model in which the dynamic is different in the congested and non-congested zone (incompressible vs. compressible dynamic). The hard congested model is the limit of the soft one as the parameter within the singular presure vanishes.
For each model, we prove the existence of traveling waves, and we study their ...

35B35 ; 35Q35 ; 35R35

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Geometric structures naturally appear in fluid motions. One of the best known examples is Saturn’s Hexagon, the huge cloud pattern at the level of Saturn’s north pole, remarkable both for the regularity of its shape and its stability during the past decades. In this paper we will address the spontaneous formation of hexagonal structures in planar viscous flows, in the classical setting of Leray’s solutions of the Navier-Stokes equations. Our analysis also makes evidence of the isotropic character of the energy density of the fluid for sufficiently localized 2D flows in the far field: it implies, in particular, that fluid particles of such flows are nowhere at rest at large distances.
Geometric structures naturally appear in fluid motions. One of the best known examples is Saturn’s Hexagon, the huge cloud pattern at the level of Saturn’s north pole, remarkable both for the regularity of its shape and its stability during the past decades. In this paper we will address the spontaneous formation of hexagonal structures in planar viscous flows, in the classical setting of Leray’s solutions of the Navier-Stokes equations. Our ...

35Q30 ; 76D05

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