P-adic numbers and ultrametric calculus

Faculty Member

Shiang Tang

Given a prime number p, the field of p-adic numbers Q_p were first introduced by Kurt Hensel in 1897, which can be thought of as the completion of the field of rationals Q with respect to the p-adic absolute value, similar to how one constructs the field of real numbers R from Q. Inside Q_p, there is a ring Z_p of p-adic integers, whose geometry resembles that of a Cantor set. P-adic numbers have many powerful applications in number theory.

In this project, we will touch upon many themes related to p-adic numbers, such as transcendental number theory and p-adic analysis (a.k.a. ultrametric calculus). We will explore p-adic analogues of well-known real and complex numbers, such as Euler's identity e^{i\pi}=-1, and Liouville numbers, which are certain types of transcendental real numbers. We will also study special functions in ultrametric calculus, such as the Artin-Hasse exponential, which is a power series emerged from the study of reciprocity laws in number theory. There are many more interesting topics and questions left to understand in the realm of p-adic numbers!

Team Meetings

weekly

Project Difficulty

Intermediate

Undergrad Prerequisites

Math 347, Math 417. Basic field theory is needed. Knowing some algebraic number theory would be a plus but is not required.