A glimpse into continuous combinatorics of posets, polytopes, and matroids

TL;DR

This paper explores continuous analogues of classical combinatorial structures like posets and polytopes, providing new formulas, dualities, and topological insights that extend finite combinatorics into a continuous setting.

Contribution

It introduces systematic methods for studying continuous versions of combinatorial objects, including new formulas, dualities, and topological results that generalize finite combinatorics.

Findings

Euler formula for continuous convex polytopes

Duality for continuous matroids

Euler characteristic of Grassmannian ideals

Abstract

This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of the `homotopy complementation formula' for topological posets and its relation to the results of Kallel and Karoui about `weighted barycenter spaces' and a conjecture of Vassiliev about simplicial resolutions of singularities. We also include an extension of the index inequality (Sarkaria's inequality) based on interpreting diagrams of spaces as continuous posets.