210514 Joost Nuiten

Differential graded Lie algebras play an important role in deformation theory as algebraic objects classifying the infinitesimal neighbourhoods of moduli spaces around a basepoint. An informal principle asserts that geometric objects without a fixed basepoint should admit a similar description in terms of curved Lie algebras, which have a `differential' whose square is controlled by a curvature element. In this talk, I will discuss the relation between two algebraic models for the formal neighbourhood of a moduli space around a manifold, rather than around a single point: in terms of dg-Lie algebroids and in terms of curved Lie algebras over the de Rham complex. In particular, I will describe an embedding of the homotopy category of dg-Lie algebroids into the homotopy category of such curved Lie algebras. Joint work with Damien Calaque and Ricardo Campos.