Moduli Spaces in Graph Theory and Comparison Theorems

A moduli space is a parameter space for classes of geometric objects of interest. A desirable property of moduli space is that nearby points in the moduli space specify nearby objects. Moduli spaces can be thought of as giving a universal space of parameters for geometric objects. The moduli space contains rich information about that object and consequently provides information to discover or construct the object being parameterized by the moduli space. The moduli space of trees and the moduli space of networks are homeomorphic to algebraic fans spanned by root subsystems of type D that arise in the moduli space of smooth marked del Pezzo surfaces. This correspondence sheds a promising light on the study of mathematical biology in terms of understanding and discovering evolutionary models. I will also briefly introduce how they are tied to the moduli space of algebraic curves and extend to the moduli space of super curves, which are algebraic curves with additional supersymmetric or supergeometric structures. I will conclude my talk with an exposition on the construction of dual graphs of SUSY curves with Neveu–Schwarz and Ramond punctures and describe how the moduli space of genus 0 SUSY graphs coincides with the aforementioned moduli space of trees.