Super Hurwitz Numbers

The first aim of this project is to construct simple examples of super Riemann surfaces using complex analysis. Classical Riemann surfaces can be constructed as branched covers of the complex plane. A branched cover of degree n can be described in elementary terms via elements of the symmetric group Sn and in particular produces many simple examples. This project will search for an analogous description of a branched cover of the super complex plane related to the classical examples. In the classical case, the enumeration of such covers is known as a Hurwitz number. The aim in the super case is to define and calculate super Hurwitz numbers.

Each Riemann surface can be represented by a hyperbolic surface. This brings a different set of techniques, such as hyperbolic trigonometry, to the study of Riemann surfaces. Similarly, each super Riemann surface can be represented by a super hyperbolic surface. This project will use this viewpoint to study the moduli space of super Riemann surfaces via its ring of functions. Simple examples of families of super hyperbolic surfaces can be constructed this way. We aim to perform super integration on such a family.

Miles Koumouris

The University of Melbourne

Miles Koumouris is a mathematics and statistics student at the University of Melbourne. His research interests include probability, stochastic processes, geometry and topology. Miles runs a successful tutoring business, and engages in collaborative problem-solving for competition and leisure. In his spare time, he enjoys playing the trumpet and coaching for We Are Tennis.

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