Math 665 Fall 2013
Total positivity

Lectures: MWF 11-12

Instructor: Thomas Lam, 2834 East Hall,

Office Hours: by appointment.

I will assume familiarity with linear algebra and this is the only prerequisite in the beginning. Towards the later half of the course, it will be helpful to be familiar with (1) Coxeter groups and root systems, and (2) semisimple complex algebraic groups.

Grading: There will be weekly-biweekly problem sets. There will be a term paper roughly 5 typed pages in length which can either (1) be expository, or (2) explore one of the open problems discussed in class. There will be no final exam.

Lecture notes: Notes for TNN part of Grassmannian (very rough)


Problem sets: Problem sets are posted here and are due in class on the stated date. Starred problems can be handed in anytime before December 6.
Some of the starred problems, particularly the open ones are suitable for the term paper. You are welcome to work in groups of 2-3 (but no bigger!) if you choose to do this.
Problem Set 1 (Due Friday September 13) Last modified: September 4 Comments
Problem Set 2 (Due Wednesday September 25) Last modified: September 13
Problem Set 3 (Due Friday October 4) Last modified: September 30 (there was a typo in Problem 2, part 2; thanks Chris for pointing this out!)
Problem Set 4 (Due Friday October 11) Last modified: October 4
Problem Set 5 (Due Friday October 25) Last modified: October 11 Solution to Problem 4 part (4)
Problem Set 6 (Due Wednesday November 6) Last modified: October 28 (A few typos were fixed thanks to Gabe)
Problem Set 7 (Due Friday November 15) Last modified: November 6

Lecture summary:

[Pin] A. Pinkus, Totally Positive Matrices, Cambridge Tracts in Mathematics
[Lus] G. Lusztig, Total positivity in reductive groups, in "Lie theory and geometry", Progress in Mathematics, 123, Birkhauser Boston 1994, 531-568.
[Kar] S. Karlin, Total positivity, Volume 1, Stanford University Press.
[FZ] S. Fomin and A. Zelevinsky, Total positivity: tests and parametrizations, The Mathematical Intelligencer, 22 (2000), 23-33.
[Bre] F. Brenti, Unimodal, Log-concave and Polya Frequency Sequences in Combinatorics, Memoirs AMS 413 (1989).
[Pos] A. Postnikov, Total positivity, Grassmannians, and networks
[LP] T. Lam and P. Pylyavskyy Total positivity in loop groups I: whirls and curls, Adv. in Math. 230 (2012), 1222-1271.
[BZ] A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), 77-128.
[Pyl] P. Pylyavskyy, Course on total positivity
[GK] A. Goncharov and R. Kenyon Dimers and cluster integrable systems
[Ker] S. Kerov, Asymptotic representation theory of the symmetric group and its applications in analysis

List of topics (tentative and potentially ambitious):
Core topics:

Possible applications: