University of Wisconsin-Madison
University of Wisconsin-Madison
Mathematics Department

Math 875
Topics in Combinatorics
Symmetric functions
Lecturer: Arun Ram

Spring 2004

Course description:

Symmetric functions are ubiquitous in all parts of mathematics -- some of the first places one meets these are in characteristic polynomials, Galois theory, invariant theory, character theory. In recent advances one sees symmetric functions playing fundamental roles in things such as Lafforgue's fields medal work the work of Neil Trudinger on partial differential equations and in the work of Okounkov-Pandharipande on Gromov-Witten invariants and the Virasoro conjecture.

In this course I will begin with the various important bases of the ring of symmetric functions and then
move on to applications. I will first do the "classical" applications of symmetric functions
in linear algebra (determinants, immanants, stochastic matrices) and in Galois theory and then ask the audience for requests.
Hopefully the audience will tell me which applications they are interested in.
If the audience has no requests I will do
(1) coinvariants
(2) the cohomology and K-theory of flag varieties
(3) the study of harmonics

Lectures: Tuesday-Thursday 1:00-2:15.

Quizzes: (pdf file)