Hurwitz Theory

This page serves as a compilation of the main takeaways of various topics in Hurwitz Theory. In particular, the following discussions and remarks are to make clear the ideas and wholistic understanding of mathematics. One of the most important things to do in math is to understand the insight that a theorem or technique can provide in your larger understanding of the field. This is what I have attempted to do here to the best of my ability.

Within each section I choose to omit certain proofs, whose proof techniques seem to "get in the way" of the main takeaways. When this is done there will be a link to a proof in the "proofs of the trade tab".

The main references for this are "Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory" by Renzo Cavalieri and "Algebraic Curves and Riemann Surfaces" by Rick Miranda. For a nice exposition of the key compositions we also refer to "Hurwitz Numbers and Monodromy Representations" by Jennifer Li

Table of Contents

Introduction to Riemann Surfaces
The Riemann-Hurwitz Formula and Holomorphic Maps of Riemann Surfaces
Homotopy, The Fundamental Group, Covering Spaces and Hurwitz Numbers
Monodromy Representations, The Symmetric Group, and Hurwitz Numbers
The Cut-and-Join Equation
Hurwitz Theory in Positive Characteristic
Tropical Hurwitz Numbers
Hurwitz Spaces and the Moduli Space of Curves
Hurwitz Numbers and String Theory

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