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UID:calendar.110201.field_date.0@www.wpi.edu
DTSTAMP:20200702T152540Z
CREATED:20180920T132938Z
DESCRIPTION:Description of Event: \n\n\n\nPhD Dissertation Proposal Present
ation by Brian Kodalen\n\n\n\nTitle: Cometric Association Schemes\n\n\n\nA
bstract: The combinatorial objects known as association schemes arise in g
roup theory\, extremal graph theory\, coding theory\, the design of experi
ments\, and even quantum information theory. One may think of a d-class as
sociation scheme as a d+1 dimensional matrix algebra closed under the entr
ywise product containing I and J. In this context\, an imprimitive scheme
is one which admits a subalgebra (subscheme) of block matrices\, also clos
ed under the entrywise product. Such systems of imprimitivity provide us w
ith quotient schemes\, smaller association schemes which are often easier
to understand\, providing useful information about the structure of the la
rger scheme. One important property of any association scheme is that we m
ay ﬁnd a basis of d + 1 idempotent matrices for our algebra. A cometric as
sociation scheme is one whose idempotent basis may be ordered E0\, E1\, …\
, Ed so that there exist polynomials q0\, q1\,…\, qd with qi ◦ (E1) = Ei a
nd deg(qi) = i for each i. Imprimitive cometric schemes relate closely to
spherical t-distance sets\, sets of unit vectors with only t distinct inn
er products\, such as equiangular lines and mutually unbiased bases. A sim
ilar type of association schemes known as metric schemes have been studied
extensively with fundamental results such as a classiﬁcation of imprimiti
ve metric schemes dating back to the early 1970’s. Analogous results for t
he cometric case weren’t settled until nearly four decades later\, with ma
ny other questions still open today.\n\n\n\nAfter introducing association
schemes with relevant terminology and deﬁnitions\, this talk focuses on im
primitive cometric association schemes\, especially those with small d. We
will introduce and examine three projects spanning the previous four year
s and select theorems from each for the purpose of illustration:\n\n\n loca
l connectivity of general association schemes\;\n linked systems of symmetr
ic designs\;\n positive semidefinite cones of cometric association schemes.
\n\n\n\n\nIn the ﬁrst case we step towards answering a conjecture about th
e connectivity of association schemes which has been open for 20 years. In
the second\, we construct the only known examples not using the Kerdock s
et parameters. Finally\, in the last project\, we ﬁnd parameter restrictio
ns independent of those already known\, resolving many open cases in onlin
e tables. We believe that these results\, and the techniques used to reach
them\, will further our understanding of the young subject of cometric as
sociation schemes.\n\n\n\nDissertation Committee:\n\n\n\nDr. William Marti
n\, WPI (Advisor)\n\n\n\nDr. Peter J. Cameron\, Queen Mary\,\n\n\n\nUniver
sity of London\n\n\n\nDr. Padraig Ó Catháin\, WPI\n\n\n\nDr. Peter Christo
pher\, WPI\n\n\n\nDr. William M. Kantor\, University of Oregon\n\n\n\nDr.
Gábor N. Sárközy\, WPI\n\n\n\n
DTSTART;TZID=America/New_York:20180926T150000
DTEND;TZID=America/New_York:20180926T170000
LAST-MODIFIED:20180920T132938Z
LOCATION:Salisbury Laboratories
SUMMARY:Mathematical Sciences - PhD Dissertation Proposal Presentation - 'C
ometric Association Schemes' by Brian Kodalen
URL;TYPE=URI:https://www.wpi.edu/news/calendar/events/mathematical-sciences
-phd-dissertation-proposal-presentation-cometric
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