Colloquium Schedule for 1996-7
Formal moduli problem
Alexander Voronov
Michigan State University and MIT
Any formal moduli problem is believed to be associated with a differentialgraded (DG) Lie algebra. Whatever object you deform, you should expect tofind a DG Lie algebra, whose first cohomology describes the space ofinfinitesimal deformations and whose moduli space of solutions of theMaurer-Cartan equation describes the moduli space in the formalneighborhood of your initial object. In this generality very limited toolsare available, and if your object is geometric, such as a complexmanifold, you would not like to disregard the rich collection of toolsoffered by geometry. A step back from an abstract DG Lie algebra in thisdirection is the tensor product of a DG commutative algebra and a Liealgebra, the DG commutative algebra indicating the presence of geometry. We will discuss some results of Beilinson and Ginzburg on the structure ofa concrete moduli space, the one of G-bundles over a Riemann surface. Wewill also report on some recent progress of Ginzburg and the speakertowards generalizing those results to the formal moduli problem associatedwith the tensor product of a DG commutative and a Lie algebras.