Configurations of points. (English) Zbl 1037.58007

Atiyah, Michael

Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 359, No. 1784, 1375-1387 (2001).

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

Reviewer: Adela-Gabriela Oiaga (Bucureşti)

MSC:

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

Keywords:

configurations; flag manifolds; symmetric group