Spring 2026
Anders Buch
Subtitle:
Linear Algebraic Groups
Course Description:
A linear algebraic group is an affine variety with a group structure; every such group is a Zariski closed subgroup of a general linear group GL(n). The theory of linear algebraic groups is analogous to the theory of compact Lie groups, but is based on algebraic geometry, which makes it possible to work over fields of positive characteristic. Any connected reductive linear algebraic group is uniquely determined by its root datum, consisting of two dual root systems embedded in dual lattices. We will discuss the ingenious methods used to carve out this information, starting from any linear algebraic group. For example, the roots are weights of the action of a maximal torus on the tangent space at the identity, and each root determines a root subgroup isomorphic to SL(2) or PSL(2), which in turn provides a coroot in the form of the maximal torus of the root subgroup. Before we get there we must first study many preliminary topics, such as the action of a group on its tangent space and coordinate ring, Jordan decomposition in arbitrary linear algebraic groups, the structure of commutative algebraic groups, quotients by closed subgroups, and the existence and properties of maximal tori and Borel subgroups, including Borel's fixed point theorem.
Text:
T. A. Springer, Linear Algebraic Groups (2nd edition)
Prerequisites:
16:640:551 (16:640:353 is recommended).
******************************************
Fall 2023
Lisa Carbone
Subtitle:
Topics in Algebra II
Course Description:
We will study infinite dimensional Lie algebras such as Kac-Moody algebras and generalized Kac-Moody algebras, also known as Borcherds algebras, as well as their connections with vertex operator algebras and physical theories.
Text:
Lecture notes will be provided
Prerequisites:
16:640:550 - Lie Algebras or equivalent
**********************************************************************************
Spring 2023
Lisa Carbone
Subtitle:
Infinite dimensional Lie algebras
Course Description:
We will study Kac--Moody algebras, Borcherds (generalized) Kac--Moody algebras and the Monster Lie algebra.
Text:
Typed notes will be provided.
Prerequisites:
A first course in Lie Algebras
**********************************************************************************
Fall 2021
Lev Borisov
Subtitle:
Topics in Algebra II: Introduction to the moduli space of curves
Course Description:
In this class, we will study complex algebraic curves and their moduli spaces. Moduli spaces reveal how objects like varieties or schemes of a particular type behave in families. The moduli space parametrizing stable n-pointed curves of genus g, gives insight into the study of smooth curves and their degenerations. As curves arise in many contexts, the moduli space of curves is a meeting ground, where constructions from diverse fields can be tested and explored.
Text:
Prerequisites:
Graduate algebra
**********************************************************************************
PLEASE NOTE: the course information changes from semester to semester for this course number. Specifics for each semester below.
Spring 2021 - James Lepowsky
Subtitle:
Topics in vertex operator algebra theory
Course Description:
Introduction to and development of selected topics and examples in vertex operator algebra theory, adapted to the interests of the students. Applications of the theory and current research topics will be highlighted. This course will be accessible to students without prior experience in vertex operator algebra theory. More specifically, one of the central themes of vertex operator algebra theory is the hierarchy: Error Correcting Codes --> Lattices --> Vertex Operator Algebras. The most important example of this hierarchy is: The Golay Code --> The Leech Lattice --> The Moonshine Module Vertex Operator Algebra, which has the Monster finite simple group as its automorphism group; which exhibits deep connections with the theory of modular functions in number theory; and which essentially forms an example of a string theory in theoretical physics. We will develop the theories of error correcting codes and lattices, and show how the remarkable properties of the Golay code and Leech lattice lead to correspondingly remarkable properties of the Moonshine Module. Along the way, we will introduce and motivate the relevant basic concepts of vertex operator algebra theory.
Text:
I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, Boston, 1988 and materials to be distributed.
Prerequisites:
Basic algebra
Previous Semesters:
- Fall 2020 Prof. Tiep
- Spring 2020 Prof. Y.Z. Huang
- Fall 2019 Prof. Sahi
- Spring 2019 Prof. Y.Z. Huang
- Spring 2018 Prof. Lepowsky