# Moduli spaces and geometric invariant theory: old and new perspectives

## TL;DR

This paper reviews classical and recent advances in constructing moduli spaces via geometric invariant theory, highlighting extensions to non-reductive groups and stacks that enable new moduli space constructions.

## Contribution

It introduces two new developments extending geometric invariant theory to non-reductive groups and stacks, broadening the scope of moduli space construction.

## Key findings

- Extended GIT to non-reductive groups
- Applied GIT to stacks for new moduli spaces
- Surveyed classical and recent progress in the field

## Abstract

Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to construct moduli spaces, and explain two new developments extending this theory to non-reductive groups and to stacks, which enable the construction of new moduli spaces.