MA 792J Symmetric Functions and Vertex Operators
Fall 2003
Instructor: Naihuan Jing, Office: 234 HA, 3-3584,
email:
Tentative Schedule, TH 11:20--12:35

Course Description:
Symmetric polynomials are widely used in many mathematical researches. The importance can be partly explained
by the fact that the subject has been associated with famous names such as Euler, Gauss, Jacobi, Cauchy, Schur,
Weyl, etc. This classical subject has become a central part of algebraic combinatorics during the last several
decades. Moreover the study of symmetric polynomials has turned out to be a multi-disciplinary field that has
 connections with algebraic geometry, invariant theory, representation theory, Lie groups and algebras,
PDE (KdV and KP hierarchy equations), statistical mechanics, random matrix models, to name a few.
In this introductory course I will try to cover the basic materials in symmetric polynomials and
some generalization (Schubert polynomials etc). We then survey the recent approach of vertex operators to
symmetric functions. Basic techniques of vertex operators will be covered to help understand this part of
vertex operator calculus. Students with maturity in linear algebra will have sufficient background for the
course. Group theory and related algebraic notions will be reviewed when needed. I will try to build the
materials from scratch and present the course in a down-to-earth fashion.

References:
There will be no formal textbooks and the materials are drawn
from the following:
R. Stanley, Enumerative combinatorics, II
Fulton and Harris, Representation Theory, a first course, Springer-Verlag, 1991
I. G. Macdonald, Symmetric functions and Hall polynomials, 1995
I. Frenkel, J. Lepowsky, A. Meurman, Vertex operator algebras and the Monstor, 1988
and research papers of myself and others.

Grading Policy:
The student is required to give a short presentation at the end of the semester.