Counting Covers: A Case Study of Problem Solving in Higher-Level Math
Category: Research Poster
Author(s): Ryan Grimm
Presenter(s): Ryan Grimm
Mentors(s): Renzo Cavalieri
The Hurwitz number counts, up to isomorphism, the number of covers of a Riemann surface of a certain genus, holding constant the ramification profile of such a cover. With this information alone, the Hurwitz number is effectively incalculable beyond simple examples. However, through translating the problem first to monodromy representations, then representation theory, Hurwitz numbers become significantly more practical computations. Examining Hurwitz numbers as a case study provides insight into much of the modern approach of mathematics: analyzing the intersections of fields to gain insight. Higher-level math today conducts research by translating problems through different fields to gain the advantages those other fields provide. This research was conducted through review of relevant literature and academic interviews with mathematics faculty. The results of this case study solidify knowledge related to Hurwitz number computations. Additionally, these results improve understanding of the expectations surrounding higher-level mathematical research and learning. Further analysis can identify how similar patterns of translating problems across fields emerge in other areas of mathematical research, and how this perspective benefits such research.