# Introduction

Today I'll be reporting on joint work with my advisor Valery Alexeev and research collaborators Philip Engel and Luca Schaffler.

Moduli arise in connection with classification problems in algebraic geometry. The basic ingredients of a classification problem are a collection of objects $A$ and an equivalence relation $\sim$ on $A$; the problem is to describe the set of equivalence classes $A / \sim$. One can usually find some discrete invariants which partition $A / \sim$ into a countable number of subsets, but in algebraic geometry this very rarely gives a complete solution of the problem. Almost always there exist "continuous families" of objects of $A$, and we would like to give $A / \sim$ some algebro-geometric structure to reflect this fact. This is the object of the theory of moduli.

The subject has its origins in the theory of elliptic functions, where one shows that there is a continuous family of such functions parameterised by the complex numbers. The word "moduli" is due to Riemann, who showed in his celebrated paper of 1857 on abelian functions that an isomorphism class of Riemann surfaces of genus $g$ is determined by $3g-3$ parameters.
However it is only very recently that one has been able to formulate moduli problems in precise terms and in some cases to obtain solutions to them.