Dates/times:

Location: Seminar Room, building VII

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.