**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

- 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.
- I will assign one or two people to each lecture.
- We will meet in my office a few days before each lecture to discuss it.
- 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 | Notes | Speaker |

9/1 | Introduction, organization, Poisson Binomial Distributions, Heilmann-Lieb Theorem | N. Srivastava | |||

9/8 | Real stable polynomials, closure properties. | N. Srivastava | |||

9/15 | Multiaffine stable polynomials, Lieb-Sokal lemma, polarization, Borcea-Branden characterization. | 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. Gurvits-Samorodnitsky. | Z. Bartha, S. Mukherjee | |

10/6 | Interlacing families, restricted invertibility. | survey sec 2-3 | blog post, Dedieu. | 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-Vinzant | Helton-Vinnikov | pdf draft | J. Kileel |

12/1 | The LLL and the Independence Polynomial | Scott-Sokal | pdf 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

- Robin Pemantle's Survey pdf.
- David Wagner's Survey pdf
- Borcea-Branden: The Lee-Yang and PĆ³lya-Schur Programs. I. Linear Operators Preserving Stability pdf.
- Borcea-Branden-Liggett: Negative Dependence and the Geometry of Polynomials. pdf.

Tentative list of topics

- Gurvits' lower bound on the permanent of a doubly stochastic matrix.
- Negative association properties of Strongly Rayleigh Measures and applications.
- The Lovasz Local Lemma via the independence polynomial.
- Existence of Ramanujan Expander Graphs.
- Mixed Characteristic Polynomials, the Kadison-Singer Problem.
- Extension of Kadison-Singer to Strongly Rayleigh Measures, the Traveling Salesman Problem.
- Connections with Free Probability.
- Hyperbolic polynomials and cones, hyperbolic programming.
- The Lax conjecture --- is every hyperbolicity cone the feasible region of a semidefinite program?
- The Helton-Vinnikov theorem --- every bivariate real stable polynomial is a determinant of PSD matrices.
- Borcea-Branden's characterization of linear operators preserving stability.

__Combinatorial and Probabilistic Applications__

__Random matrices, Expected Characteristic Polynomials, and Interlacing Families__

__Hyperbolic Polynomials__

__Characterization Theorems__

Additional topics we may cover

- The Lee-Yang theorem, phase transitions in statistical physics.
- Mixed discriminants and volumes, Alexandrov-Fenchel inequalities.