Date of Award

8-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Committee Chair

Fabrice Baudoin

Committee Member 1

Steven R. Bell

Committee Member 2

Laszlo Lempert

Committee Member 3

David B. McReynolds

Abstract

Hypoelliptic differential operators and associated geometries with origins in Lie theory are studied. We prove upper bounds on the dimension of Killing fields of analytic pseudosubriemannian manifolds under certain technical hypotheses. Existence and uniqueness results for adapted complex structures in open subsets of cotangent bundles of analytic subriemannian manifolds are proved. A generalized Lichnerowicz theorem expressing the difference between a connection laplacian and a Dirac laplacian for arbitrary linear connections and quadratic forms is proved, along with a preliminary result in local index theory for subriemannian metrics. We prove general results on the ubiquity of hypoelliptic sublaplacians arising in reductive Lie theory from the natural filtered structure of the tangent bundle of flag manifolds. A framework for studying the heat kernels of such operators from the standpoint of abstract harmonic analysis is developed involving branching the regular representation with respect to the inclusion of a closed subgroup which is transverse to the horizontal distribution of a given sublaplacian. In the compact case explicit formulæ are given.

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