This thesis would not have been completed without the invaluable assistance of my advisor, Judy Packer. Her advice and guidance helped me to persevere. I appreciate Marty Walter for acting as my second reader and providing valuable feedback. I would also like to thank Arlan Ramsay, Frederic Latremoliere, and Karl Gustafson for serving on my committee. I would also like to acknowledge Franz Luef for fruitful conversations about my thesis. I would be remiss if I did not thank Amy, Justin, Angela, Mike, and Katherine for being my friends throughout these seven years. I am a richer and better person from our many conversations and experiences, both mathematical and otherwise. I am particularly thankful for my fellow Horcrux hunters, Justin and Amy. Graduate school would not have been the same without the five of you.