Lie-infinity algebroids and singular foliations

A singular foliation is, at the name indicates, a foliation whose leaves may not be of the same dimension. The perspective on foliations adopted in this talk involves sheaves of vector fields, opening the possibility to treat foliations and their singularities from an algebraic point of view. In particular we will show how foliations are related to the Lie algebroid machinery, and introduce the notion of Lie-infinity algebroids, which are the natural generalization of both Lie-infinity algebras and Lie algebroids. We will show that under mild assumptions, most singular foliations admit a resolution, which can be equipped with a Lie-infinity algebroid structure. This structure is unique (up to homotopy) and thus possesses some universality properties, hence its name: the universal Lie-infinity algebroid associated to the singular foliation. Universal Lie-infinity algebroids may be considered as "linearizations" of singular foliations. In particular, they open the possibility to generalize to singular foliations some notions that are well defined on regular foliations. If time permit we will discuss one such example: the modular class of a singular foliation.
Zoom 875 9016 0151, for password ask V. M. Nezhinskij: nezhin@pdmi.ras.ru.