You are all expert at differentiating by x, but you might not be familiar with differentiating by 13 or another prime number.

In this talk, we will discuss how the integers, or polynomial rings over the integers, can be considered as a geometric object. Our motivating question is to find all of the “corners” or “folds” — singularities — in the solution set of equations in such geometric objects. Motivated by this problem, we will introduce Buium and Joyal’s notion of p-derivation, a version of differentiation by a prime number. Using p-derivations, we will give a description of the locus of singularities in geometric objects over the integers. We will also discuss a few other applications of p-derivations.

This is based on joint work with Melvin Hochster.