Abstract: Algebraic groups are a well studied object that arise when one has an algebraic variety with a group structure compatible with the variety. In the same vein algebraic monoids are varieties with a monoid structure imposed on them. An interesting thing happens to certain algebraic groups and algebraic monoids called the Bruhat deomposition, which provides a wealth of knowledge about the groups/monoids in terms of double cosets. We'll take a look at two collections of algebraic monoids and their Bruhat decompositions, and determine generating functions for the "sizes" of their Bruhat decompositions.

Abstract: We will use (primary and) secondary cohomology operations to describe the structure of the stable stems in low dimensions and compute a few Toda brackets in the stable stems.