Speaker
Mathieu Stienon, Penn State University
Abstract
Exponential maps arise naturally in the contexts of Lie theory
and smooth manifolds. The infinite jets of these classical exponential
maps are related to Poincaré-Birkhoff-Witt isomorphisms
and the complete symbols of differential operators. It turns out
that these formal exponential maps can be extended to the context
of graded manifolds. For dg manifolds, the formal exponential maps
need not be compatible with the homological vector field
and the incompatibility is captured by a cohomology class
reminiscent of the Atiyah class of holomorphic vector bundles.
Indeed, the space of vector fields on a dg manifold carries
a natural L∞ algebra structure whose binary bracket
is a cocycle representative of the Atiyah class of the dg manifold.
In particular, the de Rham complex associated with a foliation
carries an L∞ algebra structure akin to the L∞ algebra structure
on the Dolbeault complex of a Kähler manifold discovered
by Kapranov in his work on Rozansky-Witten invariants.