Dartmouth Math Department
Fall 2011
Beginning Monday 26 September, we will be holding a weekly workshop on computing in mathematics, Mondays 3:00-4:50. The goal is to promote the use of computing to investigate and solve problems in pure and applied math. This fall the focus will be on the computer algebra systems Sage and Magma: For details, see below.
If you wish to be on the mailing list for this workshop, please let me know: doyle@math.dartmouth.edu.
Cheers, Peter Doyle
Workshop on Computing in Mathematics Mondays 3:00-4:50 108 Kemeny
Ben Linowitz will lead sessions on Sage, Magma, and quaternion algebras. You should plan to bring a laptop to the sessions (or a friend with a laptop). Make sure your machine is able to connect to the Dartmouth Secure network. And please check beforehand that you are able to access the math department's Sage notebook server at http://sage.dartmouth.edu (only available to users on campus).
Session 1 (26 Sep 2011). Introduction to Sage. Sage is a versatile, free sytem for mathematical computing, based on the computer language Python. The goal of this session will be to introduce the Sage system, its capabilities, and its syntax. This will be accessible to a general audience, though some examples will be drawn from Math 22/24 and 31.
Session 2 (3 Oct 2011). Quadratic Number Fields. This session will be spent studying quadratic number fields and their ideals. The explicitness inherit to these fields should make this session accessible to students who have taken Math 81 (or perhaps even just Math 71).
Sessions 3,4. Algebraic number fields. In these sessions we'll talk about more general algebraic number fields, the way that rational primes factor in these extensions and the ideal class group. None of the material covered in these sessions will be taken from the Math 81 syllabus, but we won't be moving too quickly and students who have taken Math 81 should definitely get something out of attending.
Session 5. Magma and Quaternion algebras. Magma is an extrememly powerful, non-free computer algebra system. The goal of this session will be twofold. We will begin by introducing Magma. We will then define quaternion algebras and use Magma to compute examples of quaternion algebras with various interesting properties. As far as prerequisites go, exposure to algebra at the graduate level would be desirable.
Session 6 and onwards. Quaternion orders and spectral geometry. We'll define orders in quaternion algebras and study their arithmetic. The eventual goal will be to use these orders in order to produce, using magma, very nice examples of isospectral non-isometric hyperbolic surfaces. The first known example using this method is due to Marie-France Vigneras. Not only will our examples be substantially simpler than those constructed by Vigneras, but using magma we will be able to show that our examples are the best possible using Vigneras' method.