An article on "Differentiating by Prime Numbers" by Jack Jeffries in the December 2023 Notices of the AMS prominently features our own Alex Buium's work.
The article starts like this: "It is likely a fair assumption that you, the reader, are not only familiar with but even quite adept at differentiating by x. What about differentiating by 13? That certainly didn’t come up in my calculus class! From a calculus perspective, this is ridiculous: are we supposed to take a limit as 13 changes?
One notion of differentiating by 13, or any other prime number, is the notion of p-derivation discovered independently by Joyal [Joy85] and Buium [Bui96]. p-derivations have been put to use in a range of applications in algebra, number theory, and arithmetic geometry. Despite the wide range of sophisticated applications, and the fundamentally counterintuitive nature of the idea of differentiating by a number, p-derivations are elementary to define and inviting for exploration."
The article is sprinkled with other quotes on Alex Buium's work, including his `fundamental theorem of p-derivatives' (see Theorem 1) and the following paragraph "We have collected a decent set of analogues for the basics of differential calculus for p-derivations. One can ask how far this story goes, and the short answer is very far. Buium has developed extensive theories of arithmetic differential equations and arithmetic differential geometry, building analogues of the classical (nonarithmetic) versions of these respective theories with p-derivations playing the role of usual derivatives. The reader is encouraged to check out [Bui05] [Bui17] to learn more about these beautiful theories, though our story now diverges from these. Instead, we will turn our attention towards using p-derivations to give some algebraic results with geometric flavors." Finally in Other Applications a section is dedicated to Buium's work on effective bounds on rational points and p-jet spaces, which according to the author is what motivated Buium's original work on p-derivations.
Enjoy the reading!!