Permutations, braid operations, and covers of the projective line

Directed by Dr. Carl Lian

Project Description.

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This project is concerned with the combinatorics of permutations in the symmetric group $S_d$. We will investigate how certain collections of permutations are related to each other by braid operations, which turn the ordered pair $(\sigma,\tau)$ of permutations into the pair $(\tau,\tau^{-1}\sigma\tau)$. Roughly stated, our main problem of interest will be: given a tuple of permutations satisfying certain conditions, is it always possible to perform braid operations until the tuple has more desirable (in a precise way) properties?

The story of permutations and braid operations comes from a sequence of miracles linking combinatorics, group theory, topology, complex analysis, and algebraic geometry. Understanding the question above would allow us, for example, to address the following very different-looking problem: can an algebraic curve carry infinitely many $d$-sheeted covers of the projective line, ramified over 4 points? The goal of the project is two-fold: to understand the content of the second question and how it is related to the combinatorics of permutations, and to make progress on the combinatorial problem.

Required background: Students should be familiar with the basics of group theory and in particular the example of the symmetric group $S_d$, as studied in an Abstract Algebra course. Some additional exposure to topology and/or complex variables would be helpful for understanding the context of the problem, but not required. There will also be a significant programming component to the project in order to gather data and make conjectures, so either some coding experience or a willingness to pick it up quickly would be necessary.