# Programme

**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.

**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

#### References

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. *