Graduate Defense: James Alexander Byars

Thesis Information

Title: A History of the Hurwitz Problem concerning Branched Coverings

Program: Master of Science in Mathematics

Advisor: Dr. Jens Harlander, Mathematics

Committee Members: Dr. John Clemens, Mathematics; and Dr. Uwe Kaiser, Mathematics

Abstract

A branched covering f: M –> N between surfaces gives rise to a branching data recording the covering degree and the winding numbers around the branch points. Adolf Hurwitz asked if a given abstract branching data can always be realized by an actual geometric branched covering. He found obstructions to realizability in terms of permutations. 80 years later Stephen Gersten revisited the Hurwitz problem and solved it using methods from geometric group theory.

My thesis discusses the two solutions of the Hurwitz problem and points out their strengths and weaknesses.