F Nous contacter


0

Videothèque  | enregistrements trouvés : 618

O

-A +A

Sélection courante (0) : Tout sélectionner / Tout déselectionner

P Q

Quel rapport entre la forme d'un chou-fleur des côtes de Bretagne, des vaisseaux sanguins et les structures fractales ?
Quel rapport entre une maladie génétique et un fichier de musique mp3 ?
Quel rapport entre des dessins faits par Léonard de Vinci et les lois mathématiques gouvernant la forme des plantes ou la reproduction des lapins ?
Quel rapport entre la forme de la terre, le GPS de ma voiture et un vieux puits d'Egypte ?
Pourquoi les météorologues sont capables de prédire une hausse du niveau des océans dans 100 ans mais incapables de prévoir s'il va pleuvoir dans 15 jours ?
Quel rapport entre le cerveau humain et le cerveau d'un ordinateur ?
Nous répondrons à toutes ces questions via des mathématiques simples et élégantes, accessibles à tous.
Quel rapport entre la forme d'un chou-fleur des côtes de Bretagne, des vaisseaux sanguins et les structures fractales ?
Quel rapport entre une maladie génétique et un fichier de musique mp3 ?
Quel rapport entre des dessins faits par Léonard de Vinci et les lois mathématiques gouvernant la forme des plantes ou la reproduction des lapins ?
Quel rapport entre la forme de la terre, le GPS de ma voiture et un vieux puits d'Egypte ?
Pourquoi les ...

00A06 ; 00A08 ; 68-XX ; 92-XX

Multi angle  Waves and microstructures
Weinstein, Michael (Auteur de la Conférence) | CIRM (Editeur )

Multi angle  Towers of Ramanujan graphs
Li, Winnie (Auteur de la Conférence) | CIRM (Editeur )

A $d$-regular graph is Ramanujan if its nontrivial eigenvalues in absolute value are bounded by $2\sqrt{d-1}$. By means of number-theoretic methods,infinite families of Ramanujan graphs were constructed by Margulis and independently by Lubotzky-Phillips-Sarnak in 1980's for $d=q+ 1$, where q is a prime power. The existence of an infinite family of Ramanujan graphs for arbitrary d has been an open question since then. Recently Adam Marcus, Daniel Spielman and Nikhil Srivastava gave a positive answer to this question by showing that any bipartite $d$-regular Ramanujan graph has a $2$-fold cover that is also Ramanujan. In this talk we shall discuss their approach and mentionsimilarities with function field towers. A $d$-regular graph is Ramanujan if its nontrivial eigenvalues in absolute value are bounded by $2\sqrt{d-1}$. By means of number-theoretic methods,infinite families of Ramanujan graphs were constructed by Margulis and independently by Lubotzky-Phillips-Sarnak in 1980's for $d=q+ 1$, where q is a prime power. The existence of an infinite family of Ramanujan graphs for arbitrary d has been an open question since then. Recently Adam Marcus, Daniel ...

Multi angle  Towards ternary Goldbach's conjecture
Helfgott, Harald (Auteur de la Conférence) | CIRM (Editeur )

The ternary Goldbach conjecture (1742) asserts that every odd number greater than $5$ can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant $C$ satisfies the conjecture. In the years since then, there has been a succession of results reducing $C$, but only to levels much too high for a verification by computer up to $C$ to be possible $(C>10^{1300})$. (Works by Ramare and Tao have solved the corresponding problems for six and five prime numbers instead of three.) My recent work proves the conjecture. We will go over the main ideas of the proof.
ternary Goldbach conjecture - sums of primes - circle method
The ternary Goldbach conjecture (1742) asserts that every odd number greater than $5$ can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant $C$ satisfies the conjecture. In the years since then, there has been a succession of results reducing $C$, but only to levels much too high for a verification by computer up to $C$ to be ...

11P32 ; 11N35

quasi-periodic wave equation - laplacian - hamiltonian system - symplectic form - time dependance

37K55 ; 35Q55

Multi angle  Pseudo-Anosov braids are generic
Wiest, Bert (Auteur de la Conférence) | CIRM (Editeur )

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with $n\geq 3$ strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius $l$ tends to $1$ exponentially quickly as $l$ tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated ''easily'' into a rigid braid.
braid groups - Garside groups - Nielsen-Thurston classification - pseudo-Anosov - conjugacy problem
We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with $n\geq 3$ strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius $l$ tends to $1$ exponentially quickly as $l$ tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the ...

20F36 ; 20F10 ; 20F65

Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb{Z}-algebra$. For each ideal $a$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_a(R)$ has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.

13D45 ; 13F20 ; 14B15 ; 13N10 ; 13A35

The pentagram map and its analogs act on interesting and complicated spaces. The simplest of them is the classical moduli space $M_{0,n}$ of rational curves of genus $0$. These moduli spaces have a rich combinatorial structure related to the notion of "Coxeter frieze pattern" and can be understood as a "cluster manifolds". In this talk, I will explain how to describe the action of the pentagram map (and its analogs) in terms of friezes. The main goal is to understand how does this action fit with the cluster algebra structure, in particular, with the canonical (pre)symplectic form. The pentagram map and its analogs act on interesting and complicated spaces. The simplest of them is the classical moduli space $M_{0,n}$ of rational curves of genus $0$. These moduli spaces have a rich combinatorial structure related to the notion of "Coxeter frieze pattern" and can be understood as a "cluster manifolds". In this talk, I will explain how to describe the action of the pentagram map (and its analogs) in terms of friezes. The main ...

Lorsque l'on évoque Darwin et la théorie de l'évolution, on ne pense pas aux mathématiques. Pourtant dès que l'on s'intéresse aux mécanismes de la sélection naturelle, au hasard de la reproduction et au rôle des mutations, il est indispensable de les utiliser.
Après une introduction historique aux idées de Darwin sur l'évolution des espèces, nous expliquons l'impact de sa théorie et de ses réflexions sur la communauté scientifique et l'influence qu'il a eue sur la modélisation mathématique des dynamiques de population ou de la génétique des populations. Nous développons quelques exemples d'objets mathématiques, tels les processus de branchement, qui permettent de prédire le futur d'une population (son extinction, sa diversité...) ou au contraire d'en connaître le passé biologique (l'ancêtre commun d'un groupe d'individus par exemple). L'introduction du hasard dans la modélisation des questions liées à la biodiversité et à l'évolution est fondamentale. Elle permet de prendre en compte les variabilités individuelles et de mieux comprendre l'impact des facteurs écologiques et génétiques sur l'évolution des espèces.
Ces idées seront illustrées par des exemples issus de travaux récents développés entre mathématiciens et biologistes.
Lorsque l'on évoque Darwin et la théorie de l'évolution, on ne pense pas aux mathématiques. Pourtant dès que l'on s'intéresse aux mécanismes de la sélection naturelle, au hasard de la reproduction et au rôle des mutations, il est indispensable de les utiliser.
Après une introduction historique aux idées de Darwin sur l'évolution des espèces, nous expliquons l'impact de sa théorie et de ses réflexions sur la communauté scientifique et l'influence ...

00A06 ; 00A08 ; 92-XX

déchets radioactifs - modélisation mathématique

00A06 ; 93A30

discrepancy, optimal design, Latin Hypercube Sampling, computer experiment

68U07 ; 65C60 ; 62L05 ; 62K15 ; 62k20

The simulation of random heterogeneous materials is often very expensive. For instance, in a homogenization setting, the homogenized coefficient is defined from the so-called corrector function, that solves a partial differential equation set on the entire space. This is in contrast with the periodic case, where he corrector function solves an equation set on a single periodic cell. As a consequence, in the stochastic setting, the numerical approximation of the corrector function (and therefore of the homogenized coefficient) is a challenging computational task.
In practice, the corrector problem is solved on a truncated domain, and the exact homogenized coefficient is recovered only in the limit of infinitely large domains. As a consequence of this truncation, the approximated homogenized coefficient turns out to be stochastic, even though the exact homogenized coefficient is deterministic. One then has to resort to Monte-Carlo methods, in order to compute the expectation of the (approximated, apparent) homogenized coefficient within a good accuracy. Variance reduction questions thus naturally come into play, in order to increase the accuracy (e.g. reduce the size of the confidence interval) for a fixed computational cost. In this talk, we will present some variance reduction approaches to address this question.
The simulation of random heterogeneous materials is often very expensive. For instance, in a homogenization setting, the homogenized coefficient is defined from the so-called corrector function, that solves a partial differential equation set on the entire space. This is in contrast with the periodic case, where he corrector function solves an equation set on a single periodic cell. As a consequence, in the stochastic setting, the numerical ...

35B27 ; 60Hxx ; 35R60

The purpose of this presentation is to describe the basic phenomenology of the Rayleigh-Taylor instability, from its early linear phase to its late turbulent and self-similar regime. Simple experiments are performed to illustrate this phenomenology.
fluid mechanics - Rayleigh-Taylor instability - turbulence

76E17 ; 76F25 ; 76F45

shallow water - slides - incompressible fluid

76D33 ; 76L05 ; 76B15

These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that allows to prove the existence of phase transitions. Finally, we will discuss certain probabilistic representations and their consequences. These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that ...

These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that allows to prove the existence of phase transitions. Finally, we will discuss certain probabilistic representations and their consequences. These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that ...

These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that allows to prove the existence of phase transitions. Finally, we will discuss certain probabilistic representations and their consequences. These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that ...

Multi angle  On the discovery of Lagrange multipliers
Wanner, Gerhard (Auteur de la Conférence) | CIRM (Editeur )

Z