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2 y

Research talks

It is well-known that the statement "all $\aleph_1$-Aronszajn trees are special'' is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, Ben-David and Shelah proved that, assuming GCH, for every singular cardinal $\lambda$: if there exists a $\lambda^+$-Aronszajn tree, then there exists a non-special one. Furthermore:
Theorem (Ben-David and Shelah, 1986) Assume GCH and that $\lambda$ is singular cardinal. If there exists a special $\lambda^+$-Aronszajn tree, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
This suggests that following stronger statement:
Conjecture. Assume GCH and that $\lambda$ is singular cardinal.
If there exists a $\lambda^+$-Aronszajn tree,
then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.

The assumption that there exists a $\lambda^+$-Aronszajn tree is a very mild square-like hypothesis (that is, $\square(\lambda^+,\lambda)$).
In order to bloom a $\lambda$-distributive tree from it, there is a need for a toolbox, each tool taking an abstract square-like sequence and producing a sequence which is slightly better than the original one.
For this, we introduce the monoid of postprocessing functions and study how it acts on the class of abstract square sequences.
We establish that, assuming GCH, the monoid contains some very powerful functions. We also prove that the monoid is closed under various mixing operations.
This allows us to prove a theorem which is just one step away from verifying the conjecture:

Theorem 1. Assume GCH and that $\lambda$ is a singular cardinal.
If $\square(\lambda^+,<\lambda)$ holds, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
Another proof, involving a 5-steps chain of applications of postprocessing functions, is of the following theorem.

Theorem 2. Assume GCH. If $\lambda$ is a singular cardinal and $\square(\lambda^+)$ holds, then there exists a $\lambda^+$-Souslin tree which is coherent mod finite.

This is joint work with Ari Brodsky. See: http://assafrinot.com/paper/29
It is well-known that the statement "all $\aleph_1$-Aronszajn trees are special'' is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, Ben-David and Shelah proved that, assuming GCH, for every singular cardinal $\lambda$: if there exists a $\lambda^+$-Aronszajn tree, then there exists a non-special one. Furthermore:
Theorem (Ben-David and Shelah, 1986) Assume GCH and that $\lambda$ is singular ...

03E05 ; 03E65 ; 03E35 ; 05C05

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2 y

Research talks

In this talk, I will present ColDICE[1, 2], a publicly available parallel numerical solver designed to solve the Vlasov-Poisson equations in the cold case limit. The method is based on the representation of the phase-space sheet as a conforming, self-adaptive simplicial tessellation whose vertices follow the Lagrangian equations of motion. In this presentation, I will mainly focus on describing the underlying algorithm and its practical implementation, as well as showing a few practical examples demonstrating its capabilities. In this talk, I will present ColDICE[1, 2], a publicly available parallel numerical solver designed to solve the Vlasov-Poisson equations in the cold case limit. The method is based on the representation of the phase-space sheet as a conforming, self-adaptive simplicial tessellation whose vertices follow the Lagrangian equations of motion. In this presentation, I will mainly focus on describing the underlying algorithm and its practical ...

65Mxx ; 45K05 ; 65Y05 ; 76W05 ; 85A30

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2 y

Research talks

We consider bootstrap percolation on the Erdos-Renyi graph: given an initial infected set, a vertex becomes infected if it has at least $r$ infected neighbours. The graph is susceptible if there exists an initial set of size $r$ that infects the whole graph. We identify the critical threshold for susceptibility. We also analyse Bollobas's related graph-bootstrap percolation model.
Joint with Brett Kolesnik.

05C80 ; 60K35 ; 60J85 ; 82B26 ; 82B43

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Outreach

Swiss-born mathematician Nicola Kistler was the first holder of the Jean-Morlet Chair for mathematical sciences at CIRM and, in that capacity, became the first visiting researcher in residence for six months at the Centre. His stay at CIRM lasted from early February till July 2013. He set up a program of mathematical events focusing on 'Probability', with the collaboration of Véronique Gayrard, local project leader working at Marseille's Laboratoire d'Analyse, Topologie, Probabilités (ex LATP - now I2M).
CIRM - Jean-Morlet Chair on 'Probability'
Swiss-born mathematician Nicola Kistler was the first holder of the Jean-Morlet Chair for mathematical sciences at CIRM and, in that capacity, became the first visiting researcher in residence for six months at the Centre. His stay at CIRM lasted from early February till July 2013. He set up a program of mathematical events focusing on 'Probability', with the collaboration of Véronique Gayrard, local project leader working at Marseille's ...

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Research talks

I will describe a recent framework for robust shape reconstruction based on optimal transportation between measures, where the input measurements are seen as distribution of masses. In addition to robustness to defect-laden point sets (hampered with noise and outliers), this approach can reconstruct smooth closed shapes as well as piecewise smooth shapes with boundaries.

68Rxx ; 65D17 ; 65D18

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Special events;Lagrange Days

Everything is under control: mathematics optimize everyday life.
In an empirical way we are able to do many things with more or less efficiency or success. When one wants to achieve a parallel parking, consequences may sometimes be ridiculous... But when one wants to launch a rocket or plan interplanetary missions, better is to be sure of what we do.
Control theory is a branch of mathematics that allows to control, optimize and guide systems on which one can act by means of a control, like for example a car, a robot, a space shuttle, a chemical reaction or in more general a process that one aims at steering to some desired target state.
Emmanuel Trélat will overview the range of applications of that theory through several examples, sometimes funny, but also historical. He will show you that the study of simple cases of our everyday life, far from insignificant, allow to approach problems like the orbit transfer or interplanetary mission design.
control theory - optimal control - stabilization - optimization - aerospace - Lagrange points - dynamical systems - mission design
Everything is under control: mathematics optimize everyday life.
In an empirical way we are able to do many things with more or less efficiency or success. When one wants to achieve a parallel parking, consequences may sometimes be ridiculous... But when one wants to launch a rocket or plan interplanetary missions, better is to be sure of what we do.
Control theory is a branch of mathematics that allows to control, optimize and guide systems on ...

49J15 ; 93B40 ; 93B27 ; 93B50 ; 65H20 ; 90C31 ; 37N05 ; 37N35

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Special events;Lagrange Days

The most important works of the young Lagrange were two very learned memoirs on sound and its propagation. In a tour de force of mathematical analysis, he solved the relevant partial differential equations in a novel manner and he applied the solutions to a number of acoustic problems. Although Euler and d'Alembert may have been the only contemporaries who fully appreciated these memoirs, their contents anticipated much more of Fourier analysis than is usually believed. On the physical side, Lagrange properly explained the functioning of string and air-column instruments, although he did not accept harmonic analysis as we now understand it.
Lagrange - acoustics - propagation of sound - harmonic analysis - Fourier analysis - vibrating strings - organ pipes
The most important works of the young Lagrange were two very learned memoirs on sound and its propagation. In a tour de force of mathematical analysis, he solved the relevant partial differential equations in a novel manner and he applied the solutions to a number of acoustic problems. Although Euler and d'Alembert may have been the only contemporaries who fully appreciated these memoirs, their contents anticipated much more of Fourier analysis ...

01A50 ; 35-03 ; 40-03 ; 76-03

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Special events;Lagrange Days

Mathematical modeling and numerical mathematics of today is very much Lagrangian and modern automated modeling techniques lead to differential-algebraic systems. The optimal control for such systems in general cannot be obtained using the classical Euler-Lagrange approach or the maximum principle, but it is shown how this approach can be extended.
differential-algebraic equations - optimal control - Lagrangian subspace - necessary optimality conditions - Hamiltonian system - symplectic flow
Mathematical modeling and numerical mathematics of today is very much Lagrangian and modern automated modeling techniques lead to differential-algebraic systems. The optimal control for such systems in general cannot be obtained using the classical Euler-Lagrange approach or the maximum principle, but it is shown how this approach can be extended.
differential-algebraic equations - optimal control - Lagrangian subspace - necessary optimality ...

93C05 ; 93C15 ; 49K15 ; 34H05

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Research talks

I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions. The main new ingredient is a statement in additive combinatorics concerning the structure of measures whose entropy does not grow very much under convolution. If time permits I will discuss the analogous results in higher dimensions. I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close ...

28A80 ; 37A10 ; 03D99 ; 54H20

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y

Research talks

In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. These penalty functionals are crucial to force the regularized solution to conform to some notion of simplicity/low complexity. Classical priors of this kind includes sparsity, piecewise regularity and low-rank. These are natural assumptions for many applications, ranging from medical imaging to machine learning.
imaging - image processing - sparsity - convex optimization - inverse problem - super-resolution
In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. These penalty functionals are crucial to force the regularized solution to conform to some notion of simplicity/low complexity. Classical priors of this kind includes sparsity, piecewise regularity and low-rank. These are natural assumptions for many applications, ranging from medical imaging to machine ...

62H35 ; 65D18 ; 94A08 ; 68U10 ; 90C31 ; 80M50 ; 47N10

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2 y

Research talks

We review basic properties of the moment-LP and moment-SOS hierarchies for polynomial optimization and compare them. We also illustrate how to use such a methodology in two applications outside optimization. Namely :
- for approximating (as claosely as desired in a strong sens) set defined with quantifiers of the form
$R_1 =\{ x\in B : f(x,y)\leq 0 $ for all $y$ such that $(x,y) \in K \}$.
$D_1 =\{ x\in B : f(x,y)\leq 0 $ for some $y$ such that $(x,y) \in K \}$.
by a hierarchy of inner sublevel set approximations
$\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right \}\subset R_f$.
or outer sublevel set approximations
$\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right \}\supset D_f$.
for some polynomiales $(J_k)$ of increasing degree :
- for computing convex polynomial underestimators of a given polynomial $f$ on a box $B \subset R^n$.
We review basic properties of the moment-LP and moment-SOS hierarchies for polynomial optimization and compare them. We also illustrate how to use such a methodology in two applications outside optimization. Namely :
- for approximating (as claosely as desired in a strong sens) set defined with quantifiers of the form
$R_1 =\{ x\in B : f(x,y)\leq 0 $ for all $y$ such that $(x,y) \in K \}$.
$D_1 =\{ x\in B : f(x,y)\leq 0 $ for ...

44A60 ; 90C22

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y

Research talks

La décomposition par substitution des permutations permet de voir ces objets combinatoires comme des arbres. Je présenterai d'abord cette décomposition par substitution, et les arbres sous-jacents, appelés arbres de décomposition. Puis j'exposerai une méthode, complètement algorithmique et reposant sur les arbres de décomposition, qui permet de calculer des spécifications combinatoires de classes de permutations à motifs interdits. La connaissance de telles spécifications combinatoires ouvre de nouvelles perspectives pour l'étude des classes de permutations, que je présenterai en conclusion. La décomposition par substitution des permutations permet de voir ces objets combinatoires comme des arbres. Je présenterai d'abord cette décomposition par substitution, et les arbres sous-jacents, appelés arbres de décomposition. Puis j'exposerai une méthode, complètement algorithmique et reposant sur les arbres de décomposition, qui permet de calculer des spécifications combinatoires de classes de permutations à motifs interdits. La c...

68-06 ; 05A05

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Research talks

I will present results on the dynamics of horocyclic flows on the unit tangent bundle of hyperbolic surfaces, density and equidistribution properties in particular. I will focus on infinite volume hyperbolic surfaces. My aim is to show how these properties are related to dynamical properties of geodesic flows, as product structure, ergodicity, mixing, ...

37D40

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y

Research talks

Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the number field case and on a way to strengthen it assuming a height conjecture. During the second part we will focus on function fields of positive characteristic and describe a new result obtained in a joined work with Pacheco. Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the ...

14G05 ; 11G35

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Research talks

Last year, I published together with Roberto Ferretti a new version of the quantitative subspace theorem, giving a better upper bound for the number of subspaces containing the solutions of the system of inequalities involved. In my lecture, I would like to discuss this improvement, and go into some aspects of its proof.

11J13 ; 11J68

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y

Research talks

We discuss the distribution of the trace of a random matrix in the compact Lie group USp2g, with the normalized Haar measure. According to the generalized Sato-Tate conjecture, if A is an abelian variety of dimension g defined over the rationals, the sequence of traces of Frobenius in the successive reductions of A modulo primes appears to be equidistributed with respect to this distribution. If g = 2, we provide expressions for the characteristic function, the density, and the repartition function of this distribution in terms of higher transcendental functions, namely Legendre and Meijer functions. We discuss the distribution of the trace of a random matrix in the compact Lie group USp2g, with the normalized Haar measure. According to the generalized Sato-Tate conjecture, if A is an abelian variety of dimension g defined over the rationals, the sequence of traces of Frobenius in the successive reductions of A modulo primes appears to be equidistributed with respect to this distribution. If g = 2, we provide expressions for the cha...

11G05 ; 11G10 ; 14G10 ; 37C30

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y

Research talks

We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, Zhang and Maynard-Tao, respectively. We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a ...

11N05 ; 11B05

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y

Research talks

Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows are good candidates. In this conference we determine on which hyperbolic orbifolds is the geodesic flow left-handed: the answer is that yes if the surface is a sphere with three cone points, and no otherwise.
dynamical system - geodesic flow - knot - periodic orbit - global section - linking number - fibered knot
Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows ...

37C27 ; 37C15 ; 37C10 ; 57M25

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y

Research talks

The world of groups is vast and meant for wandering! During this week, I will give seven short talks describing seven groups, or class of groups, that I find fascinating. These seven talks will be independent and I will have no intention of being exhaustive (this would be silly since there are uncountably many groups, even finitely generated!). In each talk, I will introduce the hero, state one or two results, and formulate one or two conjectures. The world of groups is vast and meant for wandering! During this week, I will give seven short talks describing seven groups, or class of groups, that I find fascinating. These seven talks will be independent and I will have no intention of being exhaustive (this would be silly since there are uncountably many groups, even finitely generated!). In each talk, I will introduce the hero, state one or two results, and formulate one or two c...

57S30 ; 58D05

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y

Research talks

Tous les fournisseurs d'applications mettent actuellement en place des infrastructures "cloud". Cette nouvelle approche de l'utilisation des logiciels va complètement changer notre comportement en tant qu'utilisateurs, mais aussi en tant qu'enseignants et en tant que chercheurs.
L'objectif de cet exposé est de dégager les grands concepts scientifiques de cette évolution technologique et commerciale.
* Pourquoi le cloud aujourd'hui?
* Qu'est-ce qui a permis son émergence si rapide maintenant?
* Qu'est-ce que ça change pour l'enseignement?
* Quels sont les nouveaux défis de recherche qui sont posés?
Tous les fournisseurs d'applications mettent actuellement en place des infrastructures "cloud". Cette nouvelle approche de l'utilisation des logiciels va complètement changer notre comportement en tant qu'utilisateurs, mais aussi en tant qu'enseignants et en tant que chercheurs.
L'objectif de cet exposé est de dégager les grands concepts scientifiques de cette évolution technologique et commerciale.
* Pourquoi le cloud aujourd'hui?
* Qu'est-ce ...

68Qxx

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