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