## Special Colloquium

Kevin Tucker

Princeton University

An Introduction to F-signature

**Abstract:**While most points on an algebraic variety are smooth, understanding the singular or non-manifold points of an algebraic variety is an essential part of many investigations. In prime characteristic p > 0, the Frobenius endomorphism or p-th power map can be used to detect and characterize singularities. To that end, the F-signature is a local numerical invariant defined via Frobenius and is thought to capture subtle arithmetic and geometric information. Unfortunately, like many so-called F-invariants, explicit computations of F-signature are often extremely difficult and a complete understanding of its meaning remains elusive. In this talk, we will give an introduction to the circle of ideas surrounding F-signature together with a number of examples and conjectures. Time permitting, we will also briefly talk about the relationship between F-signature and the singularities appearing in complex birational algebraic geometry.

Tea at 4:15

Friday December 7, 2012 at 3:00 PM in SEO 636