Lecture 1. Title: Reductive Monoids and Spherical Varieties

Abstract: In this lecture, first, we will have an in-depth look at the
theory of spherical varieties; we will present the general outline of
the classification of spherical varieties. Our goal is to explain how
reductive monoids are classified as spherical varieties. (This is
based on Rittatore’s approach.) If time permits, then we will discuss
Vinberg’s enveloping semigroups and asymptotic semigroups.

Lecture 2. Title: Structure Theory of Reductive Monoids

Abstract: In this lecture, we will review the structure of a reductive
monoid. In particular, we will discuss Putcha’s cross section lattice
and the Renner monoid of a reductive monoid. Then we will have a
closer look at the geometry of J-irreducible monoids.

Lecture 3. Title: The Nilpotent Variety of a Reductive Monoid

Abstract: The main purpose of this talk is to discuss the work of
Putcha on the nilpotent elements of a reductive monoid with zero. We
will explain Putcha’s decomposition of the nilpotent variety of a
reductive monoid. In particular, we will focus on the “dual canonical
monoids”. Here, our goal is to explain how the conjugacy action on
the nilpotent variety of a dual canonical monoid is naturally linked
with certain large Schubert varieties.

Lecture 4. Title: On a Class of Solvable Affine Monoids

Abstract: This lecture will be quite different from the previous ones
in the sense that its main focus will be on a very special family of
solvable linear algebraic monoids. More precisely, we will discuss the
algebraic monoid structure of the incidence variety of a finite poset.
We will analyze their automorphism groups. We will define the notion
of a complexity of an incidence monoid, and then determine the ones
with complexity 0 and 1. If time permits, we will discuss a
generalization of this theory.