CASSE-BRIQUES

Ce travail est issu du travail de thèse inachevé de l'un des auteurs, Marie-Claire Fournier.International audienceThis manuscript proposes a mathematical model for the game named Breakout. Its systematic study reveals unexpected complexity. After a survey of general properties of the model, we study in details a special case and exhibit combinatorial difficulties arising in another case.Cet article propose une version mathématique du jeu eponyme. Son étude systématique se révèle d'une complexité surprenante.Après un survol des propriétés générales du modèle, nous étudions plus précisément un cas très particulier puis mettons en évidence les difficultés combinatoires qui apparaissent dans un autre

On the pp-adic closure of a subgroup of rational points on an Abelian variety

In 2007, B. Poonen (unpublished) studied the $p$--adic closure of a subgroup of rational points on a commutative algebraic group. More recently, J. Bella\"iche asked the same question for the special case of Abelian varieties. These problems are $p$--adic analogues of a question raised earlier by B. Mazur on the density of rational points for the real topology. For a simple Abelian variety over the field of rational numbers, we show that the actual $p$--adic rank is at least the third of the expected value

Koszul duality for PROPs

The notion of PROP models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. We prove a Koszul duality theory for PROPs generalizing the one for associative algebras and for operads.Comment: submitted to the C. R. Acad. Sci. Pari