Oberseminar Analysis und Theoretische Physik

Seminar Analysis and Theoretical Physics

Talk on Tuesday, May 17, 2022 at 3 p.m., room c311, main building; Dr. Magnus Goffeng, Lund University, The index of hypoelliptic operators on (regular) Carnot manifolds

The index of hypoelliptic operators on (regular) Carnot manifolds

We will discuss the index theory of hypoelliptic operators on Carnot manifolds – manifolds whose Lie algebra of vector fields is equipped with a filtering induced from a filtration of sub-bundles of the tangent bundle. Under the additional assumption that the Carnot manifold is regular, i.e. has isomorphic osculating Lie algebras in all fibres, and admits a flat coadjoint orbit, we provide a solution to the index problem for Heisenberg elliptic pseudodifferential operators in terms of geometric K-homology. This result extends work of Baum and van Erp on contact manifold. Up to a technical issue of constructing a global Hilbert space bundle of representations associated to the flat coadjoint orbits via Kirillov’s orbit method, the problem is reduced to computations in the K-theory of twisted groupoid C*-algebras. Examples of index theorems that follow from this solution cover Toeplitz operators and operators of the form ∆H + γT  on regular polycontact manifolds.

Joint work with Alexey Kuzmin.