Combinatorial Models of Discrete Vector Bundle Geometry

Definitions and mathematical statement

The requested objects are a discrete vector bundle, a connection on that bundle, and its curvature. Write these schematically as $E$, $\nabla$, and $F_{\nabla}$, respectively. These symbols label structures to be defined by the sought theory; the supplied record does not specify their underlying spaces, axioms, or operations.

Research problem. Develop a combinatorial model of discrete vector bundles equipped with connection and curvature, specifying the objects and operations that represent this geometry.

Unresolved scope. The original question asks for a combinatorial version of "this machinery," without supplying the full antecedent. The record does not prescribe a class of combinatorial base spaces, a coefficient field, a bundle rank, a definition of discrete connection or curvature, or criteria for correspondence with smooth geometry. It also gives no boundary conditions or quantifiers over admissible inputs. Consequently, the question is a request to develop a theory rather than a uniquely specified existence proposition; a more restrictive mathematical statement would require additional choices not supplied here.

The supplied literature assessment reports constructions of discrete vector-bundle-valued cochains, covariant exterior derivatives, wedge products, curvature, and a discrete Bianchi identity. It also states that a complete combinatorial replacement of all smooth machinery has broader scope than those constructions. Neither that broader scope nor a particular remaining theorem is specified.

Short English statement

Develop a combinatorial theory of vector bundles with connection and curvature. The source leaves the underlying combinatorial setting and required axioms unspecified. Its supplied, dated assessment labels the problem partially solved and describes existing constructions, while leaving the broader intended target unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005150$, title Combinatorial Models of Discrete Vector Bundle Geometry, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems.
• Supplied report citation: Hilbert Complexes: Analysis, Applications, and Discretizations (2023), Oberwolfach Reports, pp. 1603--1659, DOI 10.4171/owr/2022/29. https://doi.org/10.4171/owr/2022/29. The extracted passage identifies p. 1647 and the contribution Discrete Vector Bundles with Connection and Curvature, by Anil N. Hirani, joint work with Daniel Berwick-Evans and Mark Schubel. The registry separately gives the proposed year as 2022.
• Supplied literature reference: Berwick-Evans--Hirani--Schubel, discrete vector bundles. https://arxiv.org/abs/2104.10277.
• Supplied literature reference: finite element systems for vector bundles. https://doi.org/10.1007/s10208-022-09555-x.
• Supplied literature reference: OWR report. https://ems.press/content/serial-article-files/46966.
• Literature checked at: 2026-08-22. The status partially_solved and the accompanying assessment are supplied dated curation, not independently verified here. The background explicitly requests independent verification of time-sensitive status and cited claims.