BRANCHED COVERS OF CURVES WITH FIXED RAMIFICATION LOCUS

I would like to thank: • First and foremost, my advisor, David Harbater. At many points during my years as a graduate student I would enter his office having exhausted my current methods and leave with several new approaches with which to tackle a problem. His meticulous proofreading elucidated many unwritten rules concerning the syntax of mathematical writing to me. • The faculty members at Penn who served as instructors for my courses. In particular I would like to thank Ted Chinburg, Tony Pantev, and Florian Pop for teaching me the fundamentals of commutative algebra, number theory, and algebraic geometry. • My fellow graduate students at Penn, in particular those with whom I col-laborated on the first and second year coursework. Although during my first year I spent many late nights in the office, the camaraderie and willingness to explain to others what one already knew surely decreased the number of such ii nights greatly. I would especially like to thank Hilaf Hasson and Ryan Man-ion. To Hilaf, to borrow a phrase from his thesis acknowledgments, for taking the role of being my mathematical big brother seriously. To Ryan for always being willing to struggle with me to figure out the details of an argument. • The professors and teachers who instructed me in math in college and high school. Their passion made math appealing enough for me to pursue a grad-uate degree in the subject. I would especially like to thank David Barth-Hart and Likin Simon Romero of RIT and Ray Callaghan of Fargo North Hig