Victor Roca i Lucio
Abstract
The main goal of this thesis is to develop the integration theory of curved homotopy Lie algebras. In the first chapter, we develop the operadic calculus needed: we encode non-necessarily conilpotent coalgebras with operads and introduce their dual notion of an algebra over a cooperad, which we call "absolute algebras". The first sections of this chapter are a state of the art, and the last three sections are made of original results. In the second chapter, we develop the new theory of curved operadic calculus, which allows us to encode curved types of algebras with curved operads. Finally, in the third chapter, we develop the integration theory of curved absolute homotopy Lie algebras using the tools introduced in the first two chapters. This new approach allows us to obtain applications to deformation theory and rational homotopy theory.