The Milnor fiber is a topological space which can be associated to certain germs of singular complex analytic or algebraic varieties. For isolated hypersurface singularities and isolated complete intersection singularities, the Milnor fiber can be described by a single "Milnor number." Beyond these cases, the topology of the Milnor fiber is often much more complicated. An alternative generalization of the classical isolated situations is the "singular Milnor fiber," which often has a simple topology even for nonisolated singularities. We'll discuss joint work with James Damon on finding a way to compute the corresponding "singular Milnor number." In a few cases, this work has led to ways of computing information about the classical Milnor fiber.

In this talk we'll be covering some work by Sandeep Varma concerning rational submonoids of algebraic monoids. We begin by reviewing the Bruhat decomposition for algebraic groups, as discovered by Francois Bruhat. From there we'll take two different paths, the first will generalize the Bruhat decomposition to groups with BN-pairs. The second will deal with the Bruhat decomposition for algebraic monoids. Finally we will cover the results from Varma's paper which culminate in extending the Bruhat decomposition to rational submonoids.