University of WisconsinMadison

Math 875 

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 OkounkovPandharipande on GromovWitten 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 Ktheory of flag varieties
(3) the study of harmonics
Lectures: TuesdayThursday 1:002:15.
Quizzes: (pdf file)