Amnon Yekutieli, BGU

An averaging process for unipotent group actions – in differential geometry

Abstract: 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.

• slides

• notes

• compact notes (for printing)

updated: 8 Nov 2019