Date:	Mon, February 4, 2013
Time:	17:15
Place:	Research II Lecture Hall

Abstract: The real numbers, the complex numbers, the quaternions, and the octonions comprise the four composition algebras. The first three are usually seen to be the well-behaved members of the family; the octonions are, to quote John Baez, "the crazy old uncle nobody lets out of the attic."

In this broadly-accessible talk, I will let the octonions down from the attic and tell you why it should be invited to all family gatherings. In the first part of the talk, I will introduce some of its fundamental properties and describe its most important ring of integers, the integral Cayley numbers of Coxeter. In the second part, I will describe a beautiful and remarkably simple algorithm of Rehm that leads to a unique factorization theorem for this ring, despite its non-associativity. Just as remarkably, there are several elementary questions about factorization in this and related rings that remain unanswered.