Algebraic stacks

Algebraic stacks are advanced mathematical structures used to study geometric objects and their symmetries, especially when these objects have automorphisms or self-equivalences. They generalize classical spaces like varieties by allowing for the systematic handling of objects that can have different automorphism groups, capturing both geometric shape and symmetry information simultaneously. Think of them as a way to organize complex classifications—like all possible shapes of a certain type—while accounting for symmetries that might identify different forms as fundamentally related. This framework is essential in areas like algebraic geometry and number theory for understanding moduli spaces.