Institut de recherche mathématique avancée

Résumé : The variety of Lie algebras is the central example of a variety of non-associative algebras. Many properties, studied from a universal algebra point of view, were firstly introduced to Lie algebras and then generalised to other different structures. One of the many aims of categorical algebra is to give general definitions that can help to understand these notions and their generalisations. The aim of this talk is to discuss two categorical algebraic ideas, representability of actions and the existence of algebraic exponents. They are both interesting in the following way: we can characterise the variety of Lie algebras amongst all varieties of non-associative algebras over an infinite field. Moreover, these characterisations are categorical; in the sense that no elements are used, only morphisms and universal properties. We will discuss the algebraic meanings of these properties, the computational methods used to obtain the characterisations, and what can be done beyond them. Joint work with Matsvei Tsishyn, Corentin Vienne and Tim Van der Linden.

In the context of preserving stationary states, e.g. lake at rest and moving equilibria, a new formulation of the shallow water system, called Flux Globalization has been introduced by Cheng et al. (2019). This approach consists in including the integral of the source term in the global flux and reconstructing the new global flux rather than the conservative variables. The resulting scheme is able to preserve a large family of smooth and discontinuous steady-state moving equilibria. In this work, we focus on an arbitrary high order WENO Finite Volume (FV) generalization of the global flux approach. The most delicate aspect of the algorithm is the appropriate definition of the source flux (integral of the source term) and the quadrature strategy used to match it with the WENO reconstruction of the hyperbolic flux. When this construction is correctly done, one can show that the resulting WENO FV scheme admits exact discrete steady states characterized by constant global fluxes. We also show that, by an appropriate quadrature strategy for the source, we can embed exactly some particular steady states, e.g. the lake at rest for the shallow water equations. It can be shown that an exact approximation of global fluxes leads to a scheme with better convergence properties and improved solutions. The novel method has been tested and validated on classical cases and their perturbation: subcritical, supercritical and transcritical flows.

C'est un théorème célèbre de Faltings qu'il n'y qu'un nombre fini des variétés abéliennes définies sur un corps de nombre ayant des bonnes réductions en dehors d'un nombre fini des places fixées. Des résultats analogues pour les surfaces K3 ont été obtenus par Yves André, et Yiwei She. Je vais parler de mon travail récent en collaboration avec Z. Li, T. Takamatsu et H. Zou sur un analogue pour les généralisations des surfaces K3, à savoir, les variétés hyperkählériennes.

La fonction de Möbius est une fonction multiplicative sur les entiers qui apparaît en théorie des nombres comme les coefficients de l'inverse de la fonction Zeta de Riemann. On peut en fait définir la fonction de Möbius sur n'importe quel poset (ensemble partiellement ordonné). Elle peut s'interpréter comme la caractéristique d'Euler réduite d'un espace topologique naturellement associé au poset, le complexe d'ordre. D'un point de vue plus combinatoire, dans certains cas, cette fonction de Möbius peut également compter des choses intéressantes, notamment dans des configurations d'hyperplans.

The symmetric group admits two natural covering groups: the braid group and the twin group. These are obtained, respectively, by removing the involution and cubic relations in the Coxeter presentation of the symmetric group. There is a natural Burau representation for each group, which for the braid group is a q-analog of the permutation representation of the symmetric group and for the twin group is a related orthogonal representation generated by complex reflections. New instances of Schur-Weyl duality are found by examining the diagonal action of these groups on tensor powers of the Burau representation. The centralizer algebra of the action of each group is described diagramatically in terms of partial permutation and partial Brauer algebras. As a result, we obtain many representations of the braid and twin groups that can be combinatorially constructed. This is joint work with Stephen Doty.