Artin stacks

Artin stacks are sophisticated mathematical objects used to study geometric shapes that have symmetries or group actions, especially when these symmetries are complex. They generalize traditional spaces, like surfaces or curves, to include settings where points may have automorphisms, capturing the idea of "spaces" with symmetry. Think of an Artin stack as a flexible framework that allows mathematicians to analyze moduli problems—classifying and understanding families of geometric objects—by incorporating both geometric structure and symmetry information in a unified way.