4 avril 2019

Amos Nevo (Technion (Haifa))
Effective solution count in intrinsic Diophantine approximation

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Lieu : IMO, salle 2L8

Résumé : In his 1965 « Report on Diophantine approximation » Serge Lang raised the problem of establishing the approximation properties of rational points on homogeneous algebraic varieties, singling out in particular the questions of establishing Diophantine approximation exponents, an analog of Khinchin’s dichotomy theorem and an analog of W. Schmidt’s solution counting theorem.
In recent years a systematic approach to Lang’s problems has been developed for varieties homogeneous under an action of semisimple groups, and some progress towards answering the questions mentioned above has been obtained, with the answers in certain special cases being optimal. The methods involve lattice actions, ergodic theorems and spectral estimate in the automorphic representation. In the talk we will present this approach, which is based on joint work with Anish Ghosh and Alex Gorodnik.

Notes de dernières minutes : Le café culturel sera assuré à 13h par Frédéric Paulin.

Effective solution count in intrinsic Diophantine approximation  Version PDF

Rémy Degenne (LPSM)
Bandit Pure Exploration with Multiple Correct Answers

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Résumé : The active or pure exploration multi-armed bandit setting is the following : at successive times, an algorithm can sample one of K ``arms’’ (probability distributions) basing its choice on the samples observed at previous rounds. The goal of the algorithm is to answer a query, for example ``which arm has highest mean ?’’ (best arm identification) or ``is there an arm with negative mean ? if yes, return such an arm’’. A good algorithm will answer correctly with high probability and use as few samples as possible to do so.
I will present recent work on the sample complexity of active exploration in the well-studied particular case of best arm identification, and in general. In our new work with Wouter M. Koolen (see https://arxiv.org/abs/1902.03475), we determine the sample complexity of pure exploration bandit problems with multiple good answers. We derive a lower bound on the sample complexity using a game equilibrium argument. We show how continuity and convexity properties of single-answer problems ensures that an algorithm has asymptotically optimal sample complexity. However, that convexity is lost when going to the multiple-answer setting. We present a new algorithm which extends known techniques to the multiple-answer case and has asymptotic sample complexity matching the lower bound.

Bandit Pure Exploration with Multiple Correct Answers  Version PDF

Sandrine Grellier (Université d'Orléans)
Equation de Szego faiblement amortie

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Lieu : IMO, Salle 3L8

Résumé : L’équation de Szego cubique, introduite il y a une dizaine d’années comme exemple d’équation d’évolution sans dispersion, présente des phénomènes de cascades vers les hautes et les basses fréquences. Ce phénomène, que l’on peut qualifier de turbulence, est particulièrement extraordinaire pour un système pour lequel on a établi la complète intégrabilité.
Précisément, on a montré que, pour une donnée initiale $u_0$ dans un ensemble dense de $\mathcal C^\infty$, les solutions de Szego correspondantes $Z(t)u_0$ sont telles que, dans tous les espaces de Sobolev $H^s$, $s>1/2$, pour tout $M\in\mathbbR$,

$$\limsup_t\to \infty \frac| Z(t)u_0|_H^st^M=+\infty,\ \liminf_t\to \infty | Z(t)u_0|<\infty.$$

Cependant, on sait que cet ensemble dense de données initiales est d’intérieur vide !
On poursuit notre étude en introduisant un terme d’amortissement dans l’équation de Szego portant sur la plus basse fréquence et on montre que cela favorise l’existence de solutions non bornées. On démontre notamment que, pour tout $s>1/2$, il existe un ouvert non vide de données initiales dans $H^s$ qui mènent à des solutions dont la norme $H^s$ tend vers l’infini à l’infini.
Il s’agit de travaux en collaboration avec Patrick Gérard.

Equation de Szego faiblement amortie  Version PDF

Pierre-François Rodriguez (IHES)
Sign cluster geometry of the Gaussian free field

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Résumé : We consider the Gaussian free field on a large class of transient weighted graphs G, and show that its sign clusters contain an infinite connected component. In fact, we prove that the sign clusters fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs G belonging to this class include cases in which the random walk on G exhibits anomalous diffusive behavior. Our findings also imply the existence of a nontrivial percolating regime for the vacant set of random interlacements on G. Based on joint work with A. Prévost and A. Drewitz.

Sign cluster geometry of the Gaussian free field  Version PDF