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
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  
(01/13/20) No Class: ITCS 2020 
Lecture 3 (01/17/20 )  Gurvtis Machinery, LogConcavity  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)  Applications: DPPs, TSP 
 
Lecture 5 (01/27/20)  Convex Programming and Real Stable Polynomials   
Lecture 6 (01/31/20)  Hyperbolic Programming  

Lecture 7 (02/03/20)  Introduction to Hodge Theory   
Lecture 8 (02/10/20)  Combinatorial Geometries, Bergman Fan   
Lecture 9 (02/14/20)  Hodge theory and log concavity   
(02/17/20) No Class: President's Day 
Lecture 10 (02/21/20)  CLC Polynomials I  
 
Lecture 11 (02/24/20)  CLC Polynomials II: Mason's Conjecture   
Lecture 12 (02/28/20)  Expander Graphs, HDX 
 
Lecture 13 (03/02/20)  Local to Global Theorems  

Lecture 14 (03/06/20)  HDX, CLC and Matrids   
Lecture 15 (03/09/20)  Beyond LogConcavity 
 
Lecture 16 (03/13/20)  Barvinok's Interpolation Method 

