# Algebra and geometry near characteristic "p approximately zero."

## Brian Hwang, PhD

Cornell University

Tuesday, February 21, 2017

Decision Theater, 107 Farrell Hall

4:15 - 5:15 PMĀ

**Abstract:**

We usually think of algebra over rings of characteristic 0 and character p as being distinct. For example, if G is a finite group, the behavior of modules over its group ring with coefficient field k depends strongly on the characteristic of k. For example, if k is the complex numbers, indecomposable modules are irreducible and so all modules are projective, but this is not the case if k is a finite field of characteristic p dividing the order of G. Furthermore, the rings themselves are often endowed with natural "extra" structures that have no analogues in the other setting, such as Frobenius morphisms in characteristic p that do not exist in characteristic 0. So it may seem hopeless to try and find a unifying theory that fully explains the divergent behavior in both zero and nonzero characteristic. But surprisingly, some recent developments (with roots in some seemingly forgotten parts of mathematics) have been able to do precisely this! In particular, it has led to a robust theory of objects that are sort of in "intermediate" characteristic between 0 and p, or probably more accurately, in "positive characteristic p that is very close to zero." What do such objects look like geometrically? We will give some indications by explaining a simple combinatorial method for constructing geometric objects over mixed characteristic rings that yield these "tiny characteristic" objects in the limit and point towards some interesting questions raised by the existence of such a method.

ADA: Individuals requiring accommodations please contact Doreen Taylor at 802-656-3166