Connes' trace theorem on Carnot manifoldsOberseminar Global Analysis and Operator Algebras

by Ed McDonald (MI (Uni Bonn))

Tuesday Apr 21, 2026, 2:15 PM → 3:45 PM Europe/Berlin

Endenicher Allee 60/1-008 (Mathezentrum)

A Carnot manifold (also called a filtered manifold, among other
names) is a manifold $M$ whose tangent bundle $TM$ is equipped with a
nested family of sub-bundles $0=H_0,H_1,...,H_N=TM$ with the property that
the Lie bracket of a section of $H_j$ and a section of $H_k$ belongs to
$H_{j+k}$ for $j+k\leq N.$ We should think of differentiation in the
directions of $H_j$ as having ``order $j$", and the tangent spaces of $M$
can be given the structure of a nilpotent Lie group built from
$H_0,...,H_N.$ The prototypical example is a contact structure. There are
various ways to associate a pseudodifferential calculus to a Carnot
manifold, and I will describe the approach of van Erp and Yuncken based on
a modification of the tangent groupoid. I will explain some of the spectral
theory of pseudodifferential operators on a Carnot manifold, in particular
the analogy of Connes' trace theorem for this setting. Time permitting, I
will also explain some recent progress relating to operators on singular
foliations. This talk is based mostly on my paper arXiv:2601.17794.