In this course we discuss the fruitful paradigm of encoding discrete phenomena in complex multivariate polynomials, and understanding them via the interplay of the coefficients, zeros, and function values of these polynomials. For example, the states of a statistical physics model and the corresponding phase transition, the clusters in a graph, and the convex cones of linear and semidefinite programming can all be viewed in these terms. Over the last fifteen years, this perspective has led to several breakthroughs in computer science, and an unexpected bridge between distant scientific areas including combinatorics, probability, statistical physics, convex and algebraic geometry, and computer science has been built. In this course I plan to cover part of these developments with an emphasis on more recent developments specially connections to matroids, Hodge theory, high dimensional expanders and mixing time of random walks.
Administrative Information
- Instructor: Shayan Oveis Gharan
- Office Hours: By appointment, email me at shayan at cs dot washington dot edu.
- Lectures: Monday - Friday 1:30 - 2:50 at CSE II Room 271 (note change of room)
- TA: Nathan Klein, OH: Tue 2:30-3:30 at CSE2 Room 223.
Assignments
- Problem Set 1, due date Jan 27th in Canvas, solution
- Problem Set 2, due date Feb 26th in Canvas.
Related Courses
- Geometry of Polynomials in Algorithm Design by Nima Anari
- The Geometry of Polynomials in Algorithms, Combinatorics and Probability by Nikhil Srivastava
- Geometry of Polynomials and non-constructive proofs in combinatorics by Jan Vondrak
| Lecture | Topic | Notes | Bonus Materials |
|---|---|---|---|
| Lecture 1 (01/06/20) | Real Rooted Polynomials | ||
| Lecture 2 (01/10/20) | Real Stable Polynomials, Closure Properties | Pemantle Survey on Real Stability
Real Stability Testing |
|
| (01/13/20) No Class: ITCS 2020 | |||
| Lecture 3 (01/17/20 ) | Gurvtis Machinery, Log-Concavity | Approximating Mixed Volume A 2^n Approx for Permanent |
|
| (01/20/20) No Class: Martin Luther King's Day | |||
| Lecture 4 (01/24/20) | Strong Rayleigh Distributions | BBL paper on SR Dist Sampling from SR distributions | |
| Lecture 5 (01/27/20) | Maximum Entropy Convex Programs and Applications to TSP | Asymmetric TSP and Max Entropy Approximating Max Entropy |
|
| Lecture 6 (01/31/20) | Generalizing Gurvits' Machinery, Nash Welfare Maximization Problem | Generalizing Gurvits' Machinery | |
| Lecture 7 (02/03/20) | Hyperbolic Polynomials | A simple proof of Helton-Vinnikov Thm Polysize SDP representation of Hyperbolicity Cones |
|
| Lecture 8 (02/07/20) | Intro to Log Concave Polynomials | Lorentzian Polynomials Log Concave Polynomials III |
|
| Lecture 9 (02/10/20) | Log Concave Polynomials: Main Theorem | ||
| Lecture 10 (02/14/20) | Negative Correlation, Matroids and Mason's Conjecture | ||
| (02/17/20) No Class: President's Day | |||
| Lecture 11 (02/21/20) | Log Concave Polynomials and Convex Programming | Log Concave Polynomials I | |
| Lecture 12 (02/24/20) | Introduction to High Dimensional Expanders | Simpler Construction of Sparse HDX | |
| Lecture 13 (02/28/20) | Oppenheim's Trickling Down Theorem | Oppenheim's Trickling Down Thm | |
| Lecture 14 (03/02/20) | Up-Down Walks | Improved Mixing time bounds for HDX HDX and Agreement Testing" Locally Testable Codes and HDX | |
| Lecture 15 (03/06/20) | Counting Sampling and HDX | HDX and Counting Independent Sets | |
| Lecture 16 (03/09/20) | Introduction to Hodge Theory Combinatorial Geometries, Bergman Fan |
||
| Lecture 17 (03/13/20) | Hodge Theory and Log Concavity | ||
