Speaker: Richard Gonzales (Galatasaray University)
Date/Time: 27 March 2012, Tuesday, 16:00
Place: Sabanci University, FENS 2008
Title: Topological and geometric aspects of algebraic monoids
Abstract: Our first encounter with algebraic monoids is via linear algebra. The set M(n) of square matrices of size nxn is the most popular example of an algebraic monoid. Indeed, we can multiply matrices in M(n) and this operation is associative. Moreover, it has a unit , or identity matrix, and the subset of M(n) whose elements are invertible matrices forms a group: GL(n), the group of nxn-invertible matrices. However, the theory of algebraic monoids is much more complex and does not reduce to this particular example. In fact, more general and interesting classes of algebraic monoids can be obtained from linear representations of algebraic groups. Furthermore, algebraic monoids are (affine) cones over a very natural class of projective varieties: compactifications of algebraic groups. Most of the topology and geometry of these compactifications is encoded in a finite graph. In this introductory talk, I will provide an overview of the theory of algebraic monoids and explain how the topology and geometry of their associated projectivizations can be understood via combinatorial objects such as root systems, idempotents and finite semigroups.