18.704 (Seminar in Algebra and Number Theory) Course Description

There are a number of basic counting formulas in mathematics, like the formula n!/r!(n-r)! for the number of r-element subsets of an n-element set, or n! for the number of permutations of an n-element set. Some of these formulas have "q-analogues": that is, more complicated versions involving a new variable q, which reduce to the original formula when you put q=1. For example, a q-analogue of n is the polynomial

1 + q + q2 + ... + qn-1.

The subject of the seminar is ways of finding such q-analogues by doing linear algebra and geometry over finite fields. Artin's book is about linear algebra and geometry over arbitrary fields. We'll work through some parts of the book, always emphasizing the case of finite fields. (For example, the polynomial above turns out to be the number of points in an (n-1)-dimensional projective space over a field with q elements.) As a prequel, we'll look carefully at the problem of describing finite fields and calculating in them.

The details of the structure of the course will depend on enrollment, but I hope that about two-thirds of the material will be presented by the students. This might mean doing about three half-hour presentations in the course of the semester. You'll have some (but not unlimited!) control over when you speak, and the material you present.

There will be some homework problems assigned, but most of the grade will be based on participation: on your own presentations, and on your constructive criticisms of other presentations. There will be no exams or final.

David Vogan, 2-284