Title: Differential operators and Batalin-Vilkovisky structures in
noncommutative geometry

Abstract: In the first half of the talk, we recall the
Schouten-Nijenhuis bracket on the exterior algebra of vector fields =
polyvector fields (on varieties or manifolds), which is a Gerstenhaber
algebra (we recall this notion: it follows from being an exterior
algebra of a Lie algebra).  We explain a classical construction of
Koszul, which says that the variety is Calabi-Yau if and only if the
Gerstenhaber algebra structure admits an enhancement to a
Batalin-Vilkovisky (BV) structure (we will recall this notion).
Essentially, this consists of a differential operator, of square-zero,
which generates the Schouten-Nijenhuis bracket in a suitable sense.

In the second half of the talk, we introduce some basic constructions
in noncommutative geometry.  The central object of study is the space
of ``double derivations,'' which means derivations of an algebra A into
the space A \otimes A, which is considered as an A-bimodule with outer
action.  We explain that these double derivations, Der(A, A \otimes
A), have many nice properties that ordinary derivations do not: in
particular, we may form a noncommutative analogue of polyvector
fields.

We give a motivating application of double derivations: a simple way
to compute the derivative of the function e^X, where X is a *matrix*,
as a differential form on the space of matrices.

Then, we combine double derivations with the Calabi-Yau construction
from the first half of the talk, and state (imprecisely) some recent
results of Ginzburg and the speaker, generalizing Calabi-Yau geometry
to this noncommutative, ``double'' context.