# Park City lecture notes: around the inverse Galois problem

## TL;DR

This paper introduces the inverse Galois problem and explores related topics in number theory and algebraic geometry, including rationality questions, the rigidity method, and unirational varieties, based on lectures from 2022.

## Contribution

It provides an accessible overview of key concepts and recent developments in the inverse Galois problem and related research areas.

## Key findings

- Overview of the inverse Galois problem and its significance
- Discussion of Noether's problem and rationality criteria
- Introduction to the rigidity method and unirational varieties

## Abstract

The inverse Galois problem asks whether any finite group can be realised as the Galois group of a Galois extension of the rationals. This problem and its refinements have stimulated a large amount of research in number theory and algebraic geometry in the past century, ranging from Noether's problem (letting X denote the quotient of the affine space by a finite group acting linearly, when is X rational?) to the rigidity method (if X is not rational, does it at least contain interesting rational curves?) and to the arithmetic of unirational varieties (if all else fails, does X at least contain interesting rational points?). The goal of the present notes, which formed the basis for three lectures given at the Park City Mathematics Institute in August 2022, is to provide an introduction to these topics.