AWM-MAA Etta Z. Falconer Lecture 1996

Lecturer: Karen Smith, MIT and the University of Michigan

Title: Calculus mod p

Usually, derivatives are defined using “deltas and epsilons.” However, there is a purely algebraic approach to differential operators due to Grothendieck that allows us to differentiate in the abstract setting of commutative algebra or, say, functions on algebraic varieties defined over the field ?_p of p elements.
This ring is especially interesting in characteristic p. In this talk I will explain this purely algebraic approach to differential operators, with special emphasis on the case of differential operators on varieties defined over finite fields. The structure of the ring of differential operators is quite complicated in general, but in joint work with Michel Van den Bergh, we showed that if the singularities of the variety are not “too bad” the ring will be a simple ring. In this talk, I will try to explain how it is that the ring structure of differential operators may be related to singularities in algebraic geometry.