The Asymptotic Behavior of Generic Initial Systems

In the nine years that I spent with her, Grandma Campbell poured enough love and encouragement on me to last a lifetime. I will always remember her generosity, grace, and warmth. Grandma Mayes cared deeply about education and taught her family to value lifelong learning. I admire her for being a strong and tenacious woman who never swayed from her beliefs. ii ACKNOWLEDGEMENTS Karen Smith is a brilliant mathematician, an inspiring teacher, and a wonderful advisor. She has had a tremendous influence on my mathematical development and I would not be close to where I am now without her. I thank her for suggesting the main topic of this thesis, for guiding my work, and for being generous with her time. I also thank her for her encouraging words and for her plentiful advice about navigating graduate school and academia. I thank Mel Hochster for first welcoming me to Michigan and for strengthening my interest in algebra. I feel privileged to have had the opportunity to learn from such an eminent mathematician and to know such a kind and gentle man. Thank you to Daniel Erman, Victoria Booth, and Rob Lazarsfeld for their feedback on my work and for participating in my defense. I am also grateful to my many othe