Göran Gustafsson Symposium

The aim of this meeting is to bring together experts in Hodge theory and combinatorics of matroids, and the recent breakthrough interactions between the two fields.

Algebraic geometry has been recognized as a deep and powerful tool in combinatorics at least since Stanley’s proof of the g-theorem (McMullen’s conjecture), which characterizes all possible sequences counting the numbers of faces in each dimension of a simplicial polytope. Stanley’s proof, in a remarkable three-page 1980 paper, uses the Hard Lefschetz theorem for the toric variety associated to the polytope. His proof is now a prototype of an established paradigm for proving properties of integer sequences: to prove positivity, show that they are Betti numbers of some geometric object; to prove unimodality, use Hard Lefschetz; and to prove log-concavity, use the Hodge–Riemann bilinear relations from Hodge theory.

A more recent paradigm is the possibility of doing algebraic geometry without algebra and without geometry. The prototype here is the 2017 proof by Adiprasito–Huh–Katz of the Heron–Rota–Welsh conjecture on the log-concavity of the characteristic polynomial of matroids. Recall that matroids are combinatorial structures which axiomatize the notion of linear dependence, and that a matroid is said to be representable over a field F if it can be realized by a family of vectors in a vector space over F. The Heron–Rota–Welsh conjecture was known before Adiprasito–Huh–Katz for matroids representable over the complex numbers, by work of Feichtner–Yuzvinsky. But the conjecture was for an arbitrary matroid, which might not be associated to any type of geometry at all! The proof by Adiprasito–Huh–Katz builds an object from combinatorics, which ought to play the role of the cohomology ring, and proves Poincaré duality, Hard Lefschetz and the Hodge–Riemann bilinear relations for this object directly. More recently, Braden–Huh–Matherne–Proudfoot–Wang have developed a combinatorial analogue of Goresky–MacPherson’s intersection cohomology for an arbitrary matroid, and used it to prove the Dowling–Wilson top-heavy conjecture (known previously in the representable case by work of Huh–Wang) and non-negativity of coefficients of matroid Kazhdan–Lusztig polynomials (known previously in the representable case by work of Elias–Proudfoot–Wakefield). This is all very remarkable — apparently one can do abstract Hodge theory for matroids and related combinatorial objects, but we do not yet understand why.

At the forefront of all these developments is June Huh, who will deliver the plenary lecture series at the symposium. Huh’s work is characterized by a childlike curiosity and fearlessness, and has led to several completely unexpected developments. Among June Huh’s many awards and honors are the New Horizons in Mathematics prize and the Samsung Ho-Am prize. He was invited speaker at the 2018 ICM.