Math 665: Total Positivity

Professor: David E Speyer

Fall 2020

Course meets: Tuesday and Thursday, 11:30-1:00
Zoom number: 923 0365 1115 Password will be sent by e-mail; contact Professor Speyer if you wish to attend and didn't recieve an e-mail.
Office hours: Tuesday 1:30-3:00 PM and Friday 8:30-10:00 AM. Tuesday Zoom link: 925 1371 1581, Friday Zoom link: 946 4875 5047 . Password will be sent by e-mail.
Webpage: http://www.math.lsa.umich.edu/~speyer/665
Intended Level: Graduate students past the alpha algebra (593/594) courses. Students should be completely comfortable with abstract linear algebra and with combinatorial arguments.

Expected Work

Students are expected to attend the class Zoom meetings and actively participate in discussion. I am hoping to have some opportunity for collaborative work on most days, and in the final month of the term I want us to read and discuss papers. Your presence is important and valuable!

I will assign problem sets due on Fridays. I suspect these will be shorter than my usual problem sets: In the past, I have tried to assign a consistent amount of work on each problem set. This year, I am going to put problems on the problem set when I expect them to be useful or interesting for our discussion and stop when I run out of ideas for that, taking the old size of my problem sets as an upper bound. So I think these will be shorter, but I don't know.

Students will also be expected to take turns recording notes on what took place in class each day. I will assign two note takers to each date, as I think that the interactive form of class will make note taking more difficult. Notes are due 24 hours after class, and I encourage note takers to be faster.

Finally, in roughly November and December, I want to give the class over to discussion of papers on total positivity. Each student is required to select one paper to present and lead discussion of.

Homework Policy

You are welcome to consult each other provided (1) you list all people and sources who aided you, or whom you aided and (2) you write-up the solutions independently, in your own language. If you seek help from mathematicians/math students outside the course, you should be seeking general advice, not specific solutions, and must disclose this help. I am, of course, glad to provide help!

I don't intend for you to need to consult books and papers outside your notes. If you do consult such, you should be looking for better/other understanding of the definitions and concepts, not solutions to the problems.

You MAY NOT post homework problems to internet fora seeking solutions. Although I participate in some such fora, I feel that they have a major tendency to be too explicit in their help; you can read further thoughts of mine here. You may post questions asking for clarifications and alternate perspectives on concepts and results we have covered.

Problem Sets

Problem sets are due Fridays at 11:59 PM through Gradescope.

Problem Set 1 (LaTeX), due ~~Friday Sept 11~~ Monday, September 14.
Problem Set 2 (LaTeX), due ~~Friday, September 18~~ Monday, September 21.
Problem Set 3 (LaTeX), due Friday, September 25.
No problem set due Friday, October 2, due to Prof. Speyer not thinking of any good problems for it.
Problem Set 4 (LaTeX), due Friday, October 9.
No problem set due Friday, October 16, due to Prof. Speyer not thinking of any good problems for it.
Problem Set 5 (LaTeX), due Friday, October 23.
Problem Set 6 (LaTeX), due Friday, October 30.
No problem set due Friday, November 6, due to Prof. Speyer not thinking of any good problems for it.
Problem Set 7 (LaTeX), due Friday, November 13.
Problem Set 8 (LaTeX), due Friday, December 4.

Summary of class, and worksheets

Disclaimer: I didn't produce complete worksheets for this class, as I might for a more polished IBL class. I am posting what I have prepared anyway in the hope that it is useful.

September 1 — basic examples and definitions and Worksheet 1.
September 3 — the Gessel-Linström-Viennot lemma. We did the first half of Worksheet 2.
September 8 and 10 — no class due to strike.
September 15 — We described an attempt to parametrize totally positive unipotent matrices using a directed graph. Some of the material is on Worksheet 3, but a lot of things were added on the fly.
September 17 — We completed our parametrization of totally positive unipotent matrices.
September 22 — We started our discussion of the 0-Hecke monoid. Some of the motivating computations were on Worksheet 4
September 24 — We established the bijection between the 0-Hecke monoid and the symmetric group. We also described a stratification of the unipotent group by rank matrices.
September 29 — We proved that our matrix products do have the expected rank matrices, using Worksheet 5.
October 1 — We finished proving that our matrix products have the expected rank matrices, and then started discussing Bruhat decomposition, using Worksheet 6.
October 6 — We found a unique factorization form for each BwB, using Worksheet 7.
October 8 — We finished Worksheet 7 and started Worksheet 8, putting a manifold structure on B_-w B_- ∩ N₊.
October 13 — Finished Worksheet 8 and talked about the Grassmannian.
October 15 — Talked about the flag manifold and did most of Worksheet 9.
October 20 — We worked through the details of the 0-Hecke argument from last time; then we did Worksheet 10.
October 22 — We started to finish the proof, on Worksheet 11. In case you need to make examples like these, here is a snippet of Mathematica code to transform from the g coordinates to the f-coordinates.
October 27 — We finished the proof!
October 29 — We started setting up the story of total positivity in GL_n. Great class! We solved everything on Worksheet 12 and even previewed Worksheet 13.
November 3 — No class due to election day.
November 5 — Amazing class! We worked through all of Worksheets 13, 14 and 15, which finishes our study of GL_n. Next time, we start positroids.
November 10 — We started talking about Kasteleyn labelings. Progress was slow, but we got through 16.1 and 16.2 on Worksheet 16.
November 12 — We finished Worksheet 16.
November 17 — We discussed the use of matchings of planar graphs to parametrize the totally nonnegative grassmannian. We sketched proofs of the results on Worksheet 17. Despite the lack of complete proofs, we'll move on.
November 19 — We discussed cyclic rank matrices, which we will use to parameterize strata of the totally nonnegative Grassmannian. We proved Problems 18.1-18.4 on Worksheet 18. Despite the lack of a proof for 18.5, we'll move on.
December 1 — Excellent class! We finished up cyclic rank matrices on Worksheet 19 and started our final proof, that every point of the totally nonnegative Grassmannian is parametrized by a planar graph, on Worksheet 20.
December 3 — Excellent class! We worked through all of Worksheet 21. We now know that every cyclic rank matrix comes from a point of the totally nonnegative Grassmannian, and we have a clear strategy to show that every point of the totally nonnegative Grassmannian comes from a graph.
December 5 — Great end for the term! We finished the proof on Worksheet 22.

Course Notes

This course will feature day to day notes on what happened in class. Here they are!

All students will be required to take turns scribing notes for this file. When it is your turn to scribe, download the template file and write in a summary of what happened in class that day. Then e-mail it to me. The deadline for editing the update is 24 hours after the lecture. I will, in turn, proofread and edit your entries in the next 24 hours and post them back to this webpage, so that the class always has a good record of what we have covered. You are welcome to download and read the source of the notes but please do your writing in the template file; my experience is that it is easier for me to resolve merge conflicts when I copy your text into the master file than if you edit the master file directly.

If you do not know LaTeX, you should learn! I can suggest sources; I also find TeX.stackexchange incredibly useful for specific questions.

If you have forgetten when you are scheduled to scribe, you can check here.

Proposed papers for presentations

All students will be assigned to choose a paper to lead class discussion of. I'll put up more information about the structure of these presentations as the year goes on, but I am imagining something like 35 minutes of presentation, scheduled during class time, in the last two or three weeks of class.

Here are some suggested topics/papers; I am also very glad for you to suggest your own topics.

Spectral properties of totally positive matrices

Chapter 5 in Totally positive matrices by Pinkus
Interlacing inequalities for totally nonnegative matrices, Li and Matthias, Linear Algebra and its Applications (2002), and sources cited therein

Total positivity and the variation diminishing property
- Gantmakher and Krein, Sur les matrices complètement non négatives et oscillatoires, Compositio Mathematica, Volume 4 (1937), p. 445-476
- Isaac Schoenberg and Anne Whitney, A theorem on polygons in n-dimensions with applications to variation-diminishing and cyclic variation-diminishing linear transformations, Compositio Math. 9 (1951), 141–160
- Steven Karp, Sign variation, the Grassmannian, and total positivity, J. Combin. Theory Ser. A 145 (2017), 308-339.
Totally positive kernels (a version of total positivity in functional analysis). I don't know enough to know what to recommend here.
Electrical networks

Réseaux électriques planaires. I, Colin de Verdière, Comment. Math. Helv., 69 (1994),pp. 351–374
Circular planar graphs and resistor networks, Curtis, Ingerman and Morrow, Linear Algebra and its Applications, Volume 283, Issues 1–3, 1 November 1998, Pages 115-150.
Electroid varieties and a compactification of the space of electrical networks, Thomas Lam, Advances in Mathematics Volume 338, 7 November 2018, Pages 549-600.
The Space of Circular Planar Electrical Networks, Kenyon and Wilson, SIAM J. Discrete Math. 31 (2017), no. 1, 1–28.

Total positivity in other Lie groups

Total Positivity in Reductive Groups I, Lie theory and geometry, 531–568, Progr. Math., 123, Birkhäuser Boston, Boston, MA, 1994.
Total positivity in partial flag manifolds", Represent. Theory 2 (1998), 70–78
An Algebraic Cell Decomposition of the Nonnegative Part of a Flag Variety, Journal of Algebra Volume 213, Issue 1, 1 March 1999, Pages 144-154.
Total positivity for cominuscule Grassmannians, Lam and Williams, New York J. Math. 14 (2008), 53–99

Total positivity and Pfaffians. I don't know many papers about this, but it seems very natural to me to study skew-symmetric matrices with positive Pfaffians. Here are a few sources, but I also have some ideas that aren't written down.

Nonintersecting paths, Pfaffians, and plane partitions Stembridge, Adv. Math. 83 (1990), no. 1, 96–131
The Planar Ising Model and Total Positivity, Lis, Journal of Statistical Physics volume 166, pages 72–89 (2017).
Ising model and the positive orthogonal Grassmannian, Galashin and Pylyavskyy 2018 preprint (Duke to appear).

Topology of totally positive spaces
- Stratified spaces formed by totally positive varieties, Michigan Math. J. Volume 48, Issue 1 (2000), 253-270.
- Matching polytopes, toric geometry, and the totally non-negative Grassmannian, Postnikov, Speyer and Williams
- Regular cell complexes in total positivity, Hersh, Inventiones mathematicae volume 197, pages 57–114 (2014)
- Regularity theorem for totally nonnegative flag varieties, Galashin, Karp and Lam, 2019 preprint
Total positivity for Lagrangian and Symplectic Grassmanians

Total positivity for the Lagrangian Grassmannian, Karpman, Adv. in Appl. Math. 98 (2018), 25–76.
Combinatorics of symmetric plabic graphs, Karpman and Su, J. Comb. 9 (2018), no. 2, 259–278