Mathematics Ph.D. Dissertations


Some Universality and Hypercyclicity Phenomena on Smooth Manifolds

Date of Award


Document Type


Degree Name

Doctor of Philosophy (Ph.D.)


Mathematics/Mathematics (Pure)

First Advisor

Kit Chan (Committee Chair)

Second Advisor

Julia Halo (Other)

Third Advisor

Juan Bes (Committee Member)

Fourth Advisor

Mihai Staic (Committee Member)


Hypercyclicity and universality have been extensively studied in the setting of Euclidean spaces. We show how to transfer such phenomena to the setting of smooth manifolds. We focus on operators on spaces of smooth functions defined on open subsets of smooth manifolds with an emphasis on partial differentiation operators. Furthermore, we provide a framework to extend the notions of universality and hypercyclicity to spaces of smooth functions defined on general smooth and complex manifolds. Motivated by our framework, we study the hypercyclicity phenomenon on differential forms on smooth manifolds. Finally, we construct a universal sequence of integral operators on spaces of differential forms defined on open subsets of smooth manifolds.