Undergraduate Honors Theses

Thesis Defended

Spring 2016

Document Type

Thesis

Type of Thesis

Departmental Honors

Department

Mathematics

First Advisor

Katherine E. Stange

Abstract

We begin with background in algebraic number theory, specifically studying quadratic fields K and rings of integers inside those fields O_K. From there, we study properties of the matrix group PSL_2(O_K) and its congruence subgroup PSL_2(p) for some prime ideal p. The Schmidt arrangement is the orbit of the real line under Mobius transformations described by the matrices in PSL_2(O_K). We then examine how these arrangements are affected when the transformations are limited to the matrices in PSL_2(p); this forms what we will call a congruence-p subarrangement of the full arrangement. We prove some tangency properties of the congruence subarrangement and finally conjecture that the congruence-(2) subarrangement of the Schmidt arrangement of the Eisenstein integers has an Apollonian circle packing structure.

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