NSCPHD1226 Sage Days 61: Quaternion Orders and Brandt Modules

This course will show students how to work with and contribute to open-source mathematical software which they may well have to use for their entire mathematical careers. The specific subject area of this course will be on number theory and experimental mathematics. Especially, this course will focus on improving the functionality of quaternion orders and Brandt modules in sage. This is a very important topic in number theory as this allows for the efficient computation of modular forms, elliptic curves, and other arithmetic objects.

The scientific content of this masterclass will be twofold. On the computational side, the students will expand from a basic knowledge of coding to a working and contributing knowledge of an open-source piece of mathematical software which people use each day. On the mathematical side, students with a basic understanding of quaternions will learn about the integral theory of quaternion algebras and how that ties into the theory of lattices, quadratic forms, modular forms, elliptic curves, and many others.

The course will be taught in the style of a “Sage days” where each day will feature lectures by some of the leading mathematicians in this area of study and after talks all participants will write, test and document code. Homework assignments will take the form of fixing bugs, implementing algorithms or simply giving better documentation to the existing code. This gives a high level of interaction between students and other participants, whether they be Professors or other students. In particular, each student will be expected to set up an account on the “Trac” server which tracks updates to sage and to make some kind of improvement.

The mathematical prerequisites for this course will be a basic knowledge of quaternions – the first chapter of John Voight's book in preparation is recommended but a good graduate course in algebra would be sufficient. The computing prerequisites will be a basic knowledge of coding in Python – the online book “Dive into Python” is recommended. A student would also be required to bring a laptop, onto which they have installed the newest version of sage.