Polynomial-Time Point Optimization and Abrikosov Crystallization

Definitions and mathematical statement

For notation, let $d,N$ be positive integers and let $M\subseteq{\mathbb{R}}^d$ denote the prescribed submanifold. The space of configurations of $N$ distinct points on $M$ is \[ \mathcal C_N(M)=\{X\subseteq M:|X|=N\}. \] No compactness, boundary conditions, dimension of $M$, or additional geometric hypotheses are specified in the supplied record.

The record asks for the following two research objectives.

1. Construct configurations $X_N\in\mathcal C_N(M)$ in polynomial time that are optimal to high accuracy. The accompanying literature assessment describes a polynomial-time near-minimization problem associated with Smale. However, the record does not specify the objective functional, the admissible class of submanifolds, the accuracy requirement, or the computational model. Consequently, neither the approximation guarantee nor the polynomial runtime bound can be stated precisely from this record.
2. Develop a mathematical understanding of Abrikosov crystallization as $N\to\infty$, especially in two dimensions. The record does not specify an energy, a physical or geometric domain, boundary conditions, a scaling regime, the class of configurations under consideration, or the crystalline structure and mode of convergence to be established. Thus no particular crystallization theorem is determined by the supplied formulation.

These objectives preserve the supplied scope but do not constitute a fully specified mathematical conjecture. In particular, the record does not establish that both objectives concern the same energy or configuration space, and no such identification is imposed here.

Short English statement

The problem asks for efficient, highly accurate optimization of finite point configurations on a prescribed submanifold, together with an understanding of Abrikosov crystallization for large $N$, particularly in two dimensions. The objective, accuracy, computational assumptions, and crystallization model are unspecified, so a precise theorem requires further source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004636$, exact source code catalogue entry, title: Polynomial-Time Point Optimization and Abrikosov Crystallization. Project registry: data/unsolvedmath/problems.json, with additional existing records maintained in catalog/problems.json. The supplied record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is named.
• Supplied report citation: Applications of Optimal Transportation in the Natural Sciences (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/10, pp. 507--587. https://doi.org/10.4171/owr/2021/10. The record gives proposed year 2021 and an owr-year:2021 tag; these are retained alongside the supplied report citation year.
• Supplied literature source: Smale's Mathematical Problems for the Next Century. https://www.fim.uni-passau.de/fileadmin/dokumente/fakultaeten/fim/lehrstuhl/muller/SmaleProblems1998.pdf.
• Supplied literature source: Distributing many points on spheres. https://doi.org/10.1016/j.jco.2015.02.003.
• Supplied literature assessment, checked on 2026-08-22: the record labels the problem open and reports substantial asymptotic and numerical progress on Smale's polynomial-time near-minimization problem and large-system Abrikosov crystallization, while describing both as unresolved in the stated generality. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The source itself requests independent verification of time-sensitive status and cited claims.