Computational Topological Similarity of Curves

Definitions and mathematical statement

Source research question. How can topological similarity measures for curves, such as homology area and homotopy area, be applied, computed, and used efficiently in higher-dimensional settings?

Unresolved definitions and scope. The supplied record does not define homology area or homotopy area. It does not specify the ambient spaces, the class or representation of curves, boundary or endpoint conditions, or the admissible chains and homotopies. In particular, no coefficient system, geometric area functional, or convention for cases with no admissible filling or homotopy is supplied. These choices cannot be fixed from the record.

The phrase "higher-dimensional settings" does not specify whether the curves remain one-dimensional in a higher-dimensional ambient space or whether the objects being compared also have higher dimension. The requested applications, computational outputs, accuracy requirements, computational model, and meaning of efficiency are likewise unspecified. Consequently, the source supplies a research direction rather than a uniquely determined mathematical proposition or algorithmic specification. No domain, dimensional range, or universal quantifier is added in this draft.

Short English statement

The question asks how homology area, homotopy area, and related measures of curve similarity can be applied and computed efficiently in higher dimensions. Their definitions, the intended higher-dimensional objects, and the computational requirements must be specified before a precise theorem or algorithmic target can be stated.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30002955$, titled "Computational Topological Similarity of Curves," in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. No individual proposer is identified; the supplied proposed year is 2015.
• Supplied report citation: Computational Geometric and Algebraic Topology (2016), Oberwolfach Reports, pp. 2637--2699, DOI 10.4171/owr/2015/45. https://doi.org/10.4171/owr/2015/45. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. No problem-specific page or passage is supplied.
• Supplied literature source: "Oberwolfach Report 45, 2015," https://doi.org/10.4171/owr/2015/45. The report year in this label differs from the year in the supplied report citation; the discrepancy is unresolved.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and assesses it as unresolved at the stated level of generality because no single theorem or algorithmic target is specified. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field labels the statement "exact" and describes it as clear, self-contained, and publication-ready. Those are supplied editorial assertions; they do not resolve the missing mathematical definitions or establish verification against the cited report.