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CATEGORIES:Algebra Seminar
SUMMARY:Equiangular lines\, Incoherent sets and the Mathie
u Group M_23 - Neil Gillespie (Bristol)
DTSTART:20190314T150000Z
DTEND:20190314T160000Z
UID:TALK3583AT
URL:/talk/index/3583
DESCRIPTION: The problem of finding the maximum number of equi
angular lines in d-dimensional Euclidean space has
been studied extensively over the past 80 years.
The absolute upper bound on the number of equiangu
lar lines that can be found Rd is d(d+1)/2. Howeve
r\, examples of sets of lines that saturate this b
ound 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.\n\n\nTh
e known examples of equiangular lines that saturat
e the absolute bound are related to highly symmetr
ical objects\, such as the regular hexagon\, the i
cosahedron\, and the E8 and Leech lattices. By con
sidering the additional property of incoherence\,
we prove that there exists a set of equiangular li
nes that saturates the absolute bound and the inco
herence bound if and only if d=2\,3\,7 or 23.\n\n\
nWe also show that many of the maximal sets of equ
iangular lines in small dimensions can be realised
as subsets of the 276 equiangular lines in dimens
ion 23. We do this by looking at the involutions o
f the the Mathieu Group M23. In particular\, we sh
ow 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 relat
ing the E8 and Leech lattices. It also leads us to
observe a correspondence between sets of equiangu
lar lines in small dimensions and the exceptional
curves of del Pezzo surfaces\, which in turn leads
us to speculate a possible connection to the Myst
erious Duality of string theory.\n\n
LOCATION:Nuffield G13
CONTACT:Chris Parker
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