Arbeitsgemeinschaft: Combinatorial Hodge Theory

Abstract

Combinatorial Hodge theory has undergone rapid development in recent years, revealing deep connections between matroid theory, tropical geometry, toric geometry, and convex geometry. Building on the Kähler package for matroids and the emergence of Lorentzian structures across combinatorics, the field now encompasses a broad family of Hodge-theoretic phenomena arising from purely combinatorial objects. The goal of this Arbeitsgemeinschaft was to provide participants with a structured and accessible overview of these developments, emphasizing both foundational material and current research directions. The program was organized around four major themes–matroids, Hodge theory, toric methods, and Lorentzian polynomials–with lectures highlighting topics such as Baker–Bowler framework for matroids with coefficients, Chow rings of matroids and wonderful compactifications of hyperplane arrangements, Lorentzian polynomials and volume polynomials, and matroids over triangular hyperfields. Together, these lectures aimed to articulate the unifying principles underlying the subject and to prepare participants for further research in this evolving area.