MS, University of Cincinnati, 2024, Arts and Sciences: Mathematical Sciences

Much of the current literature about the Heisenberg group H is difficult for those who are in the beginning of their mathematical careers, yet H is endowed with an interesting structure which allows for the generalization of many aspects of analysis in Euclidean space. Such topics include continuity and stronger forms of the same, integral calculus, restrictions and extensions of functions, and Taylor's theorem. The goal of this thesis is to make more accessible a combination of these tenets and others, through examining a Whitney extension theorem in H. We start by building the fundamentals in a more familiar setting, namely in Euclidean 3-space. We then discuss H and its properties, including the notion of horizontality of curves in H. The concept of horizontality provides a natural segue to a version of Whitney's extension theorem for horizontal curves in H; we discuss the necessity and sufficiency of three criteria a curve in Hn must satisfy in order to have a smooth horizontal extension. We conclude by examining two other types of extension theorems, namely Lipschitz maps on metric spaces and continuous maps on normal topological spaces.

Committee: Nageswari Shanmugalingam Ph.D. (Committee Member); Gareth Speight Ph.D. (Committee Chair) Subjects: Mathematics