Hurwitz theory is a beautiful algebro-geometric theory that studies maps of Riemann Surfaces. Despite being (relatively) unsophisticated, it is typically unapproachable at the undergraduate level because it ties together several branches of mathematics that are commonly treated separately. This book intends to present Hurwitz theory to an undergraduate audience, paying special attention to the connections between algebra, geometry and complex analysis that it brings about. We illustrate this point by giving an overview of the material in the book.

Hurwitz theory is the enumerative study of analytic functions between Riemann Surfaces – complex compact manifolds of dimension one. A Hurwitz number counts the number of such functions when the appropriate set of discrete invariants is fixed. This has its origin in the 1800s in the work of Riemann, who first had the insight that multi-valued inverses of complex analytic functions can be naturally seen as functions defined on a domain which is locally, but not globally, identifiable with the complex plane: i.e. a Riemann Surface.

Studying analytic functions defined on Riemann Surfaces leads to the geometry of oriented topological surfaces, which Riemann Surfaces are. The local behavior of functions reveals a high degree of structure: analytic functions are ramified coverings; that is, coverings except at a discrete set of points where a phenomenon called ramification occurs.

Ramified coverings naturally give rise to monodromy representations, which are homomorphisms from the fundamental group of the punctured target surface to a symmetric group. The ramification at the preimages of a point b in the base is captured by the cycle type of the permutation associated with a small loop winding around the point b.

The count of all such representations can be identified with a coefficient of a specific product of vectors in the class algebra of the symmetric group: with a vector space which has a basis indexed by conjugacy classes. Elements of this basis are given by formal sums of all permutations in the same conjugacy class. A commutative multiplication is then defined by extending the group operation of the symmetric group by bilinearity.