CSE 599: Polynomial Paradigm in Algorithms Algorithms

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

Related Courses

Lecture Topic Notes Bonus Materials
Lecture 1 (01/06/20) Real Rooted Polynomials pdf
Lecture 2 (01/10/20) Real Stable Polynomials, Closure Properties pdf Pemantle Survey on Real Stability
Real Stability Testing
(01/13/20) No Class: ITCS 2020
Lecture 3 (01/17/20 ) Gurvtis Machinery, Log-Concavity pdf 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 pdf BBL paper on SR Dist
Sampling from SR distributions
Lecture 5 (01/27/20) Maximum Entropy Convex Programs and Applications to TSP pdf Asymmetric TSP and Max Entropy
Approximating Max Entropy
Lecture 6 (01/31/20) Generalizing Gurvits' Machinery, Nash Welfare Maximization Problem pdf Generalizing Gurvits' Machinery
Lecture 7 (02/03/20) Hyperbolic Polynomials pdf A simple proof of Helton-Vinnikov Thm
Polysize SDP representation of Hyperbolicity Cones
Lecture 8 (02/07/20) Intro to Log Concave Polynomials pdf 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 pdf
(02/17/20) No Class: President's Day
Lecture 11 (02/21/20) Log Concave Polynomials and Convex Programming pdf Log Concave Polynomials I
Lecture 12 (02/24/20) Introduction to High Dimensional Expanders pdf 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