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Documents : Multi angle  Conférences Vidéo | enregistrements trouvés : 686

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We discuss classical realizability, a branch of mathematical logic that investigates the computational content of mathematical proofs by establishing a correspondence between proofs and programs. Research in this field has led to the development of highly technical constructions generalizing the method of forcing in set theory. In particular, models of realizability are models of ZF, and forcing models are special cases of realizability models.

03E70 ; 03F50 ; 03F55

Multi angle  Tracing the dark matter web
Shandarin, Sergei (Auteur de la Conférence) | CIRM (Editeur )

Dark matter (DM) constitutes almost 85% of all mass able to cluster into gravitationally bound objects. Thus it has played the determining role in the origin and evolution of the structure in the universe often referred to as the Cosmic Web. The dark matter component of the Cosmic Web or simply the Dark Matter Web is considerably easier to understand theoretically than the baryonic component of the web if one assumes that DM interacts only gravitationally. One of the major differences between the DM and baryonic webs consists in the multi stream structure of the DM web. Thus it allows to use three diagnostic fields that do not present in the baryonic web: the number of streams field in Eulerian space, the number of flip flops field in Lagrangian space, and the caustic structure in the both. Although these characteristics have been known for a long time their systematic studies as fields started only a few years ago. I will report new recent results of numerical studies of the three fields mentioned above and also discuss the features of the DM web they have unveil. Dark matter (DM) constitutes almost 85% of all mass able to cluster into gravitationally bound objects. Thus it has played the determining role in the origin and evolution of the structure in the universe often referred to as the Cosmic Web. The dark matter component of the Cosmic Web or simply the Dark Matter Web is considerably easier to understand theoretically than the baryonic component of the web if one assumes that DM interacts only ...

85A15 ; 85A25 ; 85A40 ; 83F05

We discuss joint work with Doug Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider the Neumann boundary value problem for the operator $L = divA\nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. The eigenfunctions of $L$ are often localized, as a result of disorder of the potential $V$, the matrix of coefficients $A$, irregularities of the boundary, or all of the above. In earlier work, Filoche and Mayboroda introduced the function $u$ solving $Lu = 1$, and showed numerically that it strongly reflects this localization. In this talk, we deepen the connection between the eigenfunctions and this landscape function $u$ by proving that its reciprocal $1/u$ acts as an effective potential. The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum. We discuss joint work with Doug Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider the Neumann boundary value problem for the operator $L = divA\nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. The eigenfunctions of $L$ are often localized, as a result of disorder of the potential $V$, the matrix of coefficients $A$, irregularities of the boundary, or all of the above. In ...

47A75 ; 81Vxx ; 81Q10 ; 35P20

The Aldous-Broder algorithm allows one to sample the uniform spanning tree of a finite graph as the set of first-entry edges of a simple random walk. In this talk, I will discuss how this can be extended to infinite transient graphs by replacing the random walk with the random interlacement process. I will then outline how this new sampling algorithm can be used to compute critical exponents for the uniform spanning forest of $Z^d$.

60D05 ; 05C05 ; 20F65

Recent work has clarified how various natural second-order set-theoretic principles, such as those concerned with class forcing or with proper class games, fit into a new robust hierarchy of second-order set theories between Gödel-Bernays GBC set theory and Kelley-Morse KM set theory and beyond. For example, the principle of clopen determinacy for proper class games is exactly equivalent to the principle of elementary transfinite recursion ETR, strictly between GBC and GBC+$\Pi^1_1$-comprehension; open determinacy for class games, in contrast, is strictly stronger; meanwhile, the class forcing theorem, asserting that every class forcing notion admits corresponding forcing relations, is strictly weaker, and is exactly equivalent to the fragment $\text{ETR}_{\text{Ord}}$ and to numerous other natural principles. What is emerging is a higher set-theoretic analogue of the familiar reverse mathematics of second-order number theory. Recent work has clarified how various natural second-order set-theoretic principles, such as those concerned with class forcing or with proper class games, fit into a new robust hierarchy of second-order set theories between Gödel-Bernays GBC set theory and Kelley-Morse KM set theory and beyond. For example, the principle of clopen determinacy for proper class games is exactly equivalent to the principle of elementary transfinite recursion ETR, ...

03E60 ; 03E30 ; 03C62

Multi angle  Frozen and near-critical percolation
van den Berg, Jacob (Auteur de la Conférence) | CIRM (Editeur )

Motivated by solgel transitions, David Aldous (2000) introduced and analysed a fascinating dynamic percolation model on a tree where clusters stop growing ('freeze') as soon as they become infinite.
In this talk I will discuss recent (and ongoing) work, with Demeter Kiss and Pierre Nolin, on processes of similar flavour on planar lattices. We focus on the problem whether or not the giant (i.e. 'frozen') clusters occupy a negligible fraction of space. Accurate results for near-critical percolation play an important role in the solution of this problem.
I will also present a version of the model which can be interpreted as a sensor/communication network.
Motivated by solgel transitions, David Aldous (2000) introduced and analysed a fascinating dynamic percolation model on a tree where clusters stop growing ('freeze') as soon as they become infinite.
In this talk I will discuss recent (and ongoing) work, with Demeter Kiss and Pierre Nolin, on processes of similar flavour on planar lattices. We focus on the problem whether or not the giant (i.e. 'frozen') clusters occupy a negligible fraction of ...

60K35 ; 82B43

We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size N. In our model, the spread of each strain is described by the stochastic logistic SIS epidemic process in the absence of the other strain, and we assume that there is perfect cross-immunity between the two virus strains, that is, individuals infected by one strain are temporarily immune to re-infections and infections by the other strain. For the case where one strain has a strictly larger basic reproductive ratio than the other, and the stronger strain on its own is supercritical (that is, its basic reproductive ratio is larger than 1), we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population, that is, its extinction time. We further consider what happens when the difference between the two reproductive ratios may tend to 0.
This is joint work with Fabio Lopes.
We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size N. In our model, the spread of each strain is described by the stochastic logistic SIS epidemic process in the absence of the other strain, and we assume that there is perfect cross-immunity between the two virus strains, that is, individuals infected by one strain are temporarily immune to re...

60J27 ; 92D30

Gyrokinetic simulation is considered to be an essential tool to study turbulent transport driven by micro-scale instabilities in tokamak plasmas. It is roughly categorized into two approaches; delta-$f$ local and full-$f$ global approaches. In full-$f$ approach, both turbulent transport and profile evolutions are solved self-consistently under the power balance between external heat source/sink. In this talk, we address (A) numerical technique to treat such full-$f$ gyrokinetic Vlasov-Poisson equations [1] and (B) characteristics of global ion-scale turbulence and transport barrier [2]. We also discuss (C) the role of stable modes in collisionless or weakly collisional plasmas [3]. Gyrokinetic simulation is considered to be an essential tool to study turbulent transport driven by micro-scale instabilities in tokamak plasmas. It is roughly categorized into two approaches; delta-$f$ local and full-$f$ global approaches. In full-$f$ approach, both turbulent transport and profile evolutions are solved self-consistently under the power balance between external heat source/sink. In this talk, we address (A) numerical technique ...

76X05 ; 65Mxx ; 76F10 ; 82D10

In this talk I will review the recent developments on weighted distances in scale free random graphs as well as highlight key techniques used in the proofs. We consider graph models where the degree distribution follows a power-law such that the empirical variance of the degrees is infinite, such as the configuration model, geometric inhomogeneous random graphs, or scale free percolation. Once the graph is created according to the model definition, we assign i.i.d. positive edge weights to existing edges, and we are interested in the proper scaling and asymptotic distribution of weighted distances.
In the infinite variance degree regime, a dichotomy can be observed in all these graph models: the edge weight distributions form two classes, explosive vs conservative weight distributions. When a distribution falls into the explosive class, typical distances converge in distribution to proper random variables. While, when a distribution falls into the conservative class, distances tend to infinity with the model size, according to a formula that captures the doubly-logarithmic graph distances as well as the precise behaviour of the distribution of edge-weights around the origin. An integrability condition decides into which class a given distribution falls.
This is joint work with Adriaans, Baroni, van der Hofstad, and Lodewijks.
In this talk I will review the recent developments on weighted distances in scale free random graphs as well as highlight key techniques used in the proofs. We consider graph models where the degree distribution follows a power-law such that the empirical variance of the degrees is infinite, such as the configuration model, geometric inhomogeneous random graphs, or scale free percolation. Once the graph is created according to the model ...

05C80 ; 90B15 ; 60C05 ; 60D05

We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond [1, 3]. First we establish the convergence towards the Euler equation under several assumptions on the energy and on the norms of the initial data. Then we analyze the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density of particles with a moving point charge, characterized by a Dirac mass in the phase-space. We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond [1, 3]. First we establish the convergence towards the Euler equation under several assumptions on the energy and on the norms of the initial data. Then we analyze the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density of particles with a moving point charge, characterized by ...

76X05 ; 82C21 ; 35Q35 ; 35Q83 ; 35Q60 ; 82D10

Multi angle  The quantum Vlasov equation
Mauser, Norbert (Auteur de la Conférence) | CIRM (Editeur )

We present the Quantum Vlasov or Wigner equation as a "phase space" presentation of quantum mechanics that is close to the classical Vlasov equation, but where the "distribution function" $w(x,v,t)$ will in general have also negative values.
We discuss the relation to the classical Vlasov equation in the semi-classical asymptotics of small Planck's constant, for the linear case [2] and for the nonlinear case where we couple the quantum Vlasov equation to the Poisson equation [4, 3, 5] and [1].
Recently, in some sort of "inverse semiclassical limit" the numerical concept of solving Schrödinger-Poisson as an approximation of Vlasov-Poisson attracted attention in cosmology, which opens a link to the "smoothed Schrödinger/Wigner numerics" of Athanassoulis et al. (e.g. [6]).
We present the Quantum Vlasov or Wigner equation as a "phase space" presentation of quantum mechanics that is close to the classical Vlasov equation, but where the "distribution function" $w(x,v,t)$ will in general have also negative values.
We discuss the relation to the classical Vlasov equation in the semi-classical asymptotics of small Planck's constant, for the linear case [2] and for the nonlinear case where we couple the quantum Vlasov ...

35Q40 ; 35J10 ; 81Q20 ; 81S30

N. Hindman, I. Leader and D. Strauss proved that if $2^{\aleph_0}<\aleph_\omega$ then there is a finite colouring of $\mathbb{R}$ so that no infinite sumset $X+X$ is monochromatic. Now, we prove a consistency result in the other direction: we show that consistently relative to a measurable cardinal for any $c:\mathbb{R}\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb{R}$ so that $c\upharpoonright X+X$ is constant. The goal of this presentation is to discuss the motivation, ideas and difficulties involving this result, as well as the open problems around the topic. Joint work with P. Komj‡th, I. Leader, P. Russell, S. Shelah and Z. Vidny‡nszky. N. Hindman, I. Leader and D. Strauss proved that if $2^{\aleph_0}<\aleph_\omega$ then there is a finite colouring of $\mathbb{R}$ so that no infinite sumset $X+X$ is monochromatic. Now, we prove a consistency result in the other direction: we show that consistently relative to a measurable cardinal for any $c:\mathbb{R}\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb{R}$ so that $c\upharpoonright X+X$ is constant. The goal of this ...

03E02 ; 03E35 ; 05D10

Lagrange - history of mathematics - 19th century - fluid mechanics

01A55 ; 70H03 ; 76M30 ; 76B15

Multi angle  When he was one hundred years old!
Verdier, Norbert (Auteur de la Conférence) | CIRM (Editeur )

In this talk we will don't speak about Joseph-Louis Lagrange (1736-1813) but about Lagrange's reception at the nineteenth Century. "Who read Lagrange at this Times?", "Why and How?", "What does it mean being a mathematician or doing mathematics at this Century" are some of the questions of our conference. We will give some elements of answers and the case Lagrange will be a pretext in order to explain what are doing historians of mathematics: searching archives and ­ thanks to a methodology ­ trying to understand, read and write the Past.
Lagrange - mathematical press - complete works - bibliographic index of mathematical sciences (1894-1912) - Liouville - Boussinesq - Terquem
In this talk we will don't speak about Joseph-Louis Lagrange (1736-1813) but about Lagrange's reception at the nineteenth Century. "Who read Lagrange at this Times?", "Why and How?", "What does it mean being a mathematician or doing mathematics at this Century" are some of the questions of our conference. We will give some elements of answers and the case Lagrange will be a pretext in order to explain what are doing historians of mathematics: ...

01A50 ; 01A55 ; 01A70 ; 01A74 ; 01A80

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

Multi angle  Lagrange and water waves
Saut, Jean-Claude (Auteur de la Conférence) | CIRM (Editeur )

Lagrange - 19th century - water waves

01A55 ; 76B15

Multi angle  Billiard and rigid rotation
Treschev, Dmitry (Auteur de la Conférence) | CIRM (Editeur )

Can a billiard map be locally conjugated to a rigid rotation? We prove that the answer to this question is positive in the category of formal series. We also present numerical evidence that for "good" rotation angles the answer is also positive in an analytic category.
billiard systems # integrable Hamiltonian systems # normal form convergence # small divisors # elliptic fixed point # analytic conjugacy

37D50 ; 70H06

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

37K55 ; 35Q55

I will discuss some recent results with Aaron Brown and Zhiren Wang on actions by higher rank lattices on nilmanifolds. I will present the result in the simplest case possible, $SL(n,Z)$ acting on $Tn$, and try to present the ideas of the proof. The result imply existence of invariant measures for $SL(n,Z)$ actions on $Tn$ with standard homotopy data as well as global rigidity of Ansosov actions on infranilmanifolds and existence of semiconjugacies without assumption on existence of invariant measure. I will discuss some recent results with Aaron Brown and Zhiren Wang on actions by higher rank lattices on nilmanifolds. I will present the result in the simplest case possible, $SL(n,Z)$ acting on $Tn$, and try to present the ideas of the proof. The result imply existence of invariant measures for $SL(n,Z)$ actions on $Tn$ with standard homotopy data as well as global rigidity of Ansosov actions on infranilmanifolds and existence of sem...

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