Andrey Lazarev (University of Leicester)
al179@leicester.ac.uk
A lot of modern mathematics and theoretical physics revolves around the equation d(x)=½[x,x]. It is called the Maurer-Cartan (MC) equation, also the master equation, sometimes with the adjective `classical' or `quantum'. Differential geometers recognize solutions to this equation as flat connections, algebraic topologists as twisting cochains. Various notions of homotopy invariant structures (such as A-infinity algebras) admits interpretations as solutions to the master equation. (Almost) any deformation problem in characteristic zero can be expressed in terms of solutions of a certain master equation. Moreover, this equation also underpins rational homotopy theory, although this point of view is not too widespread.
In this talk I will give an abstract and unifying treatment of the MC equation. Given a nilpotent differential graded Lie algebra L one constructs a certain simplicial set MC(L), the MC space of L whose homology and homotopy admits a natural interpretation in terms of L. If L is simply-connected this correspondence is equivalent to the Quillen version of rational homotopy theory. If, however, L is not simply-connected and, better yet, is not connected then the space MC(L) exhibits rather unexpected properties.
This is joint work with M. Markl.