Berry–Robbins problem

Berry and Robbins posed the problem in 1997: construct, for every $n$, a continuous symmetry-compatible assignment from configurations of points in three-dimensional space to the corresponding flag manifold. Atiyah gave a positive solution in 2000.

Known results

• Atiyah, 2000: the required continuous $S_n$-equivariant map $C_n(\mathbb{R}^3)\to U(n)/T^n$ exists.
• Later literature distinguishes this established existence theorem from Atiyah’s more explicit candidate maps, whose validity depends on stronger linear-independence conjectures.

2019–2024 clarification

Subsequent papers repeatedly confirm Atiyah’s original solution while studying explicit constructions and the stronger Atiyah–Sutcliffe conjectures. The 2024 work proves conditional consequences for those stronger conjectures, not a new resolution or challenge to the Berry–Robbins problem.

Current status (as of August 2026): The original Berry–Robbins existence problem is settled positively by Atiyah; only related explicit-construction and stronger-conjecture questions remain open.