Celebratio Mathematica — Atiyah

Works authored by Atiyah

Works connected to Paul Sutcliffe

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M. Atiyah and P. Sutcliffe: “The geometry of point particles,” R. Soc. Lond. Proc. Ser. A 458 : 2021 (2002), pp. 1089–1115. MR 1902577 Zbl 1010.58015 article

There is a very natural map from the configuration space of \( n \) distinct points in Euclidean 3-space into the flag manifold \( U(n)/U(1)^n \), which is compatible with the action of the symmetric group. The map is well defined for all configurations of points provided a certain conjecture holds, for which we provide numerical evidence. We propose some additional conjectures, which imply the first, and test these numerically. Motivated by the above map, we define a geometrical multi-particle energy function and compute the energy-minimizing configurations for up to 32 particles. These configurations comprise the vertices of polyhedral structures that are dual to those found in a number of complicated physical theories, such as Skyrmions and fullerenes. Comparisons with 2- and 3-particle energy functions are made. The planar restriction and the generalization to hyperbolic 3-space are also investigated.

M. Atiyah and P. Sutcliffe: “Polyhedra in physics, chemistry and geometry,” Milan J. Math. 71 : 1 (2003), pp. 33–58. MR 2120915 Zbl 1050.52002 article