Equiangular lines, Incoherent sets and the Mathieu Group M_23

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• Neil Gillespie (Bristol)
• Thursday 14 March 2019, 15:00-16:00
• Nuffield G13.

If you have a question about this talk, please contact Chris Parker.

The problem of finding the maximum number of equiangular lines in d-dimensional Euclidean space has been studied extensively over the past 80 years. The absolute upper bound on the number of equiangular lines that can be found Rd is d(d+1)/2. However, examples of sets of lines that saturate this bound are only known to exist in dimensions d=2,3,7 or 23, and it is an open question whether this bound is achieved in any other dimension.

The known examples of equiangular lines that saturate the absolute bound are related to highly symmetrical objects, such as the regular hexagon, the icosahedron, and the E8 and Leech lattices. By considering the additional property of incoherence, we prove that there exists a set of equiangular lines that saturates the absolute bound and the incoherence bound if and only if d=2,3,7 or 23.

We also show that many of the maximal sets of equiangular lines in small dimensions can be realised as subsets of the 276 equiangular lines in dimension 23. We do this by looking at the involutions of the the Mathieu Group M23 . In particular, we show how the involution structure of M23 can be used to describe the roots of E8. This has the effect of providing what we believe is a new way of relating the E8 and Leech lattices. It also leads us to observe a correspondence between sets of equiangular lines in small dimensions and the exceptional curves of del Pezzo surfaces, which in turn leads us to speculate a possible connection to the Mysterious Duality of string theory.

This talk is part of the Algebra Seminar series.

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• Algebra Seminar
• Nuffield G13
• School of Mathematics Events

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