Classification of Reductive Algebraic Monoids

A reductive monoid is a Zariski-closed monoid whose group of units is a reductive group. These objects live at a nexus of several important areas of mathematics. They relate to semigroup theory: reductive monoids are exactly those algebraic monoids that are regular as semigroups. They relate to algebraic geometry: monoids are simpler geometrically than groups, and this geometry is heavily dependent on a particular torus embedding. And in many ways, the theory of reductive monoids parallels that of reductive groups.
We will discuss a beautiful combinatorial classification by Renner of normal reductive monoids with dimension-1 center. These monoids can be classified by ''polyhedral root systems'': root systems along with a rational polyhedral cone generated by characters of the torus. This combinatorial construction is an analogue of Chevalley's classification of semisimple groups, and has applications both to reductive monoid morphisms, and to their multiplicative structure.