Singularity theory is the study of how far regular local rings are from being smooth. In general, even if a ring fails to be smooth, it can still have many nice properties that allow arguments to go through: reduced, normal, CM, Gorenstein, among others.

However, there are some types of singularities in complex geometry whose definitions break down in positive characteristic. They may rely on analytic techniques, or secretly divide by n!. In positive characteristic we then need different methods to define them.

We'll talk about how these properties are defined in terms of the Frobenius map, why this map contains information about singularities, and how one can meaningfully compare singularity types in characteristic zero or positive characteristic.