An averaging process for unipotent group actions – in differential geometry

The usual weighted average of points $(z_0, ..., z_q)$ in the real vector space $R^n$, with weights $(w_0, ..., w_q)$, is translation invariant. Hence it can be seen as an average of points in a torsor Z over the Lie group $G = R^n$ (A $G$-torsor is a $G$-manifold with a simply transitive action.)

In this talk I will explain how this averaging process can be generalized to a torsor Z over a unipotent Lie group $G$. (In differential geometry, a unipotent group is a simply connected nilpotent Lie group. $R^n$ is an abelian unipotent group.)

I will explain how to construct the unipotent weighted average, and discuss its properties (functoriality, symmetry and simpliciality). If time permits, I will talk about torsors over a base manifold, and families of sections parametrized by simplices. I will indicate how I came about this idea, while working on a problem in deformation quantization.

Such an averaging process exists only for unipotent groups. For instance, it does not exist for a torus $G$ (an abelian Lie group that’s not simply connected). In algebraic geometry the unipotent averaging has arithmetic significance, but this is not visible in differential geometry.

Notes for the talk can be founds here: https://www.math.bgu.ac.il/~amyekut/lectures/average-diff-geom/abstract.html