Hodge theory in combinatorics

Abstract:

If $G$ is a finite graph, a proper coloring of $G$ is a way to color the vertices of the graph using $n$ colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if $G$ is planar, then there is at least one proper coloring of $G$ with four colors.) By a classical result of Birkhoff, the number of proper colorings of $G$ with $n$ colors is a polynomial in $n$, called the chromatic polynomial of $G$. Read conjectured in 1968 that for any graph $G$, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read’s conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh’s result was subsequently refined and generalized by Huh and Katz (also in 2012), again using substantial doses of algebraic geometry. Both papers in fact establish log-concavity of the coefficients, which is stronger than unimodality.

The breakthroughs of the Huh and Huh–Katz papers left open the more general Rota–Welsh conjecture, where graphs are generalized to (not necessarily representable) matroids, and the chromatic polynomial of a graph is replaced by the characteristic polynomial of a matroid. The Huh and Huh–Katz techniques are not applicable in this level of generality, since there is no underlying algebraic geometry to which to relate the problem. But in 2015 Adiprasito, Huh, and Katz announced a proof of the Rota–Welsh conjecture based on a novel approach motivated by but not making use of any results from algebraic geometry. The authors first prove that the Rota–Welsh conjecture would follow from combinatorial analogues of the hard Lefschetz theorem and Hodge–Riemann relations in algebraic geometry. They then implement an elaborate inductive procedure to prove the combinatorial hard Lefschetz theorem and Hodge–Riemann relations using purely combinatorial arguments.

We will survey these developments.