Petter Brändén
Geometry and combinatorics of hyperbolic polynomials

Recently methods using hyperbolic and stable polynomials have seen several spectacular applications in combinatorics, computer science, probability theory and other areas. Hyperbolic and stable polynomials are generalizations of univariate real-rooted polynomials as well as multivariate determinantal polynomials. I will give an introduction to the theory of stable and hyperbolic polynomials and give/discuss applications such as  the existence of infinite families of Ramanujan graphs of each degree, the Kadison-Singer problem and the van der Waerden conjecture. 

Exercises for Brändén's lectures


June Huh
Hodge theory in geometry, algebra, and combinatorics

I will give a broad overview of the Hard Lefschetz theorems and the Hodge-Riemann relations in the theory of polytopes, complex manifolds, reflection groups, algebraic and tropical varieties, in a down-to-earth way. Several applications to the elementary combinatorics of graphs and matroids will be introduced.

Exercises for Huh's lectures


Nicholas Proudfoot
The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials

Kazhdan-Lusztig-Stanley polynomials are general combinatorial gadgets that include, as special cases, classical Kazhdan-Lusztig polynomials, toric g-polynomials, and Kazhdan-Lusztig polynomials of matroids.  In each of these cases, many of the polynomials can be realized as intersection cohomology Poincare polynomials of certain algebraic varieties.  I will describe a general geometric framework for providing this kind of cohomological interpretation of Kazhdan-Lusztig-Stanley polynomials, with an emphasis on the examples coming from matroids.  (No previous familiarity with intersection cohomology will be assumed.)  

I will also discuss the conjectural log concavity and real-rootedness of matroidal Kazhdan-Lusztig polynomials and Z-polynomials.  Almost nothing has been proved in general, so this will consist mostly of working through some very concrete examples.

Exercises for Proudfoot's lectures



P. Brändén. Lecture notes written for Interlacing families,

P. Brändén. Geometry of zeros and applications

Wagner, David G. Multivariate stable polynomials: theory and applications. Bull. Amer. Math. Soc. (N.S.) 48 (2011), no. 1, 53–84.

Marcus, Adam W.; Spielman, Daniel A.; Srivastava, Nikhil Interlacing families I: Bipartite Ramanujan graphs of all degrees. Ann. of Math. (2) 182 (2015), no. 1, 307–325.

Marcus, Adam W.; Spielman, Daniel A.; Srivastava, Nikhil Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem. Ann. of Math. (2) 182 (2015), no. 1, 327–350.

June Huh, Benjamin Schröter, and Botong Wang. Correation bounds for fields and matroids. arXiv:1806.02675
Antoine Chambert-Loir. Relations de Hodge-Riemann et matroïdes, d’après Adiprasito, Huh et Katz. Séminaire Bourbaki , Mars 2018.
Matt Baker. Hodge theory in combinatorics. Bulletin of the American Mathematical Society 55 (2018).
Karim Adiprasito, June Huh, and Eric Katz. Hodge theory for combinatorial geometries. Annals of Mathematics 188 (2018).
June Huh and Botong Wang. Enumeration of points, lines, planes, etc. Acta Mathematica 218 (2017).

Nick Proudfoot. The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials

Published Nov. 24, 2017 3:26 PM - Last modified Aug. 28, 2019 10:30 AM