Course Descriptions

16:640:555 - Selected Topics in Algebra

Fall 2021 - Angela Gibney

Subtitle:

Topics in Algebra II: Introduction to the moduli space of curves

Course Description:

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. In this class, we will study the moduli space of curves, with a focus on its cycle theory, discussing some of the historical highlights and the important open problems.

Text:

Moduli of Curves by Joe Harris and Ian Morrison

Prerequisites:

Graduate algebra

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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: