Abstract

Someone told me once that Gaussian elimination is the one thing a mathematician learns as an undergrad and then never really uses in research. This talk is about proving this is false. I will first introduce a homological algebra version of Gaussian elimination, due to Bar-Natan, which can be used to simplify complexes (e.g., chain complexes) arising in many areas of mathematics such as algebraic topology and representation theory. Then, following Sköldberg, I will explain how to interpret this operation as a "collapse" in the sense of Forman's discrete Morse theory. Finally I will describe an application of this method to Rouquier complexes, certain interesting objects upgrading the braid group to a category.