Noncommutative moduli spaces

The purpose of this talk will be to define noncommutative
analogues of the operads made up of the Deligne-Mumford
compactification of the moduli spaces of genus 0 curves with marked points,
the little disks and the framed little disks respectively. To this end,
we use toric varieties associated to Loday realization of Stasheff
polytopes. We will show that all the properties of the classical
case are satisfied on this new noncommutative level. For instance,
we will mention applications to noncommutative differential operators
and deformation theory. [This is a joint work with Vladimir
Dotsenko and Sergey Shadrin]