## Configurations of points.(English)Zbl 1037.58007

The author discusses a problem in elementary geometry that arises from the investigations of M. V. Berry and J. M. Robbins [Proc. R. Soc. Lond., Ser. A {453}, 1771–1790 (1997; Zbl 0892.46084)], namely the following. Can one construct a continuous map from the configuration space of $$n$$ distinct particles in 3-space to the flag manifold of the unitary group $$U(n)$$?
The first non-trivial case is for $$n=2$$. An obvious solution is given.
Some results in this respect were previously obtained by the same author [In: Surveys in differential geometry, Cambridge, MA: International Press Vol. 7, 1–15 (2000)]. In particular, there is a version in which $$U(n)$$ is replaced by an arbitrary compact Lie group. This problem was treated by using an integrable system of ordinary differential equations, called Nahm’s equations, which arises from the self-dual Yang-Mills equations.
Numerical computations are given and some generalizations are discussed

### MSC:

 58D29 Moduli problems for topological structures 55R80 Discriminantal varieties and configuration spaces in algebraic topology

### Keywords:

configurations; flag manifolds; symmetric group

Zbl 0892.46084
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