Math 270: The Geometry of Polynomials in Algorithms, Combinatorics, and Probability.

Meetings: Tuesdays 3:30pm-5pm, 736 Evans Hall.

Prerequisites: elementary linear algebra, probability, and complex analysis.

Course Description: The goal of this course is to understand real-rooted polynomials and their multivariate generalizations (real stable and hyperbolic polynomials) in the context of questions in combinatorics, probability, and algorithms. We will do this by surveying some recent representative results and discussing open problems. The desired outcome is for participants to become comfortable with these objects, make connections between the areas we discuss, and potentially apply these techniques in their own research.

Format: I will give the first 2-4 lectures. The remaining lectures will be given by students who are taking the class for credit (perhaps in pairs, depending on the class size). Each student will give one lecture and scribe another lecture. The lectures will be assigned (at least two weeks in advance) and prepared as follows

  1. I will post summaries, references, and tentative plans for the topics listed below sometime in the first two weeks, and send an email soliciting preferences.
  2. I will assign one or two people to each lecture.
  3. We will meet in my office a few days before each lecture to discuss it.
  4. The speaker(s) will write up lecture notes and send them to me at least a day before the lecture.

Lecture Schedule

Date Topic Readings Additional References NotesSpeaker
9/1 Introduction, organization, Poisson Binomial Distributions, Heilmann-Lieb Theorem pdf N. Srivastava
9/8 Real stable polynomials, closure properties. pdf N. Srivastava
9/15 Multiaffine stable polynomials, Lieb-Sokal lemma, polarization, Borcea-Branden characterization. pdf N. Srivastava
9/22 Strongly Rayleigh measures and negative association. Chapter 6 of
N. Srivastava
9/29 Gurvits's Lowerbound on the Permanent, Vishnoi's application to TSP. Barvinok's notes
(sec 2-5)
Gurvits, Vishnoi.
pdf Z. Bartha, S. Mukherjee
10/6 Interlacing families, restricted invertibility. survey sec 2-3blog post, Dedieu. pdf N. Ryder
10/13 Ramanujan graphs from the matching polynomial. MSS-1 Bilu-Linial, HLW survey, talk A. Ramachandran
10/20 The Kadison-Singer Problem MSS-2 Tao's blog
blog post, Dan's talk
pdf draft A. Schild
10/26 Barrier Arguments MSS-2 Tao's blog, pdf draft A. Rusciano
11/3 KS for Strongly Rayleigh Measures Anari-Oveis Gharan Anari-Oveis Gharan II
Shayan's Course
pdf draft
Open Problems
R. Zhang
11/10 Hyperbolic polynomials and hyperbolicity cones Branden's Notes pdf draft A. El Alaoui
11/17 Hyperbolic polynomials, interlacers, and sums of squares Kummer-Plaumann-Vinzant pdf draft N. Anari
11/24 Hyperbolic Polynomials in 3 variables are determinants of PSD matrices Plaumann-VinzantHelton-Vinnikov pdf draft J. Kileel
12/1 The LLL and the Independence Polynomial Scott-Sokalpdf draft J. Liu, Y. Zhang
12/8 The Lee-Yang Theorem Piyush Srivastava's notes
Borcea-Branden sec 8
pdf draft S. Mukherjee

Basic References

Tentative list of topics

Additional topics we may cover