## UNF Graduate Theses and Dissertations

2020

Summer

Master's Thesis

#### College

College of Arts and Sciences

#### Degree Name

Master of Science in Mathematical Sciences (MS)

#### Department

Mathematics & Statistics

#### NACO controlled Corporate Body

University of North Florida. Department of Mathematics and Statistics

Daniel Dreibelbis

Raimundo Araujo Dos Santos

Denis Bell

#### Department Chair

Richard Patterson

#### Abstract

We study a problem at the intersection of harmonic morphisms and real analytic Milnor fibrations. Baird and Ou establish that a harmonic morphism from G: \mathbb{R}^m \setminus V_G \rightarrow \mathbb{R}^n\setminus \{0\} defined by homogeneous polynomials of order p retracts to a harmonic morphism \psi|: S^{m-1} \setminus K_\epsilon \rightarrow S^{n-1} that induces a Milnor fibration over the sphere. In seeking to relax the homogeneity assumption on the map G, we determine that the only harmonic morphism $\varphi: \mathbb{R}^m \setminus V_G \rightarrow S^{m-1}\K_\epsilon$ that preserves \arg G is radial projection. Due to this limitation, we confirm Baird and Ou's result, yet establish further that in fact only homogeneous polynomial harmonic morphisms retract to harmonic-morphism Milnor maps over the sphere.

COinS