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CATEGORIES:Category Theory Seminar
SUMMARY:On model theory\, noncommutative geometry and topo
i - Boris Zilber\, University of Oxford
DTSTART;TZID=Europe/London:20110201T141500
DTEND;TZID=Europe/London:20110201T151500
UID:TALK29695AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/29695
DESCRIPTION:I will start with a model theoretic notion. Zarisk
i geometries is a class of structures discovered t
o answer a classification problem. The prototypica
l version of a Zariski geometry is an algebraic va
riety over an algebraically closed field with rela
tions on it given by Zariski closed subsets. The i
nitial expectation that Zariski geometries are ess
entially of this kind were proven true to some ext
ent but have generally been overturned by new exam
ples (Hrushovski & Zilber\, 1993).\n\nThe present-
day interpretation of "new" Zariski geometries lea
ds to Noncommutative geometry. For a large class o
f noncommutative algebras\, e.g. quantum algebras
at roots of unity\, we established a duality betwe
en the algebras and Zariski geometries as their "c
o-ordinate algebras"\, typically noncommutative\,
extending the well-known duality between classical
geometric objects and the algebras of regular (co
ntinuous) functions on them. Zariski geometry in
this construction appears essentially as a categor
y of\nrepresentations of the algebra. This can be
extended to a broader geometric context\, with top
ology richer than Zariski one.\n\nA different moti
vation led the physicist C.Isham and the philosoph
er J.Butterfield to suggest a certain kind of topo
i as a possible "geometric spaces" for noncommutat
ive "co-ordinate algebras". This has been\ninvesti
gated in depth by A.Doering (Oxford). As it turned
out the two approaches have a lot in common.\n\nI
will report on a recent progress in understanding
these connections.
LOCATION:MR3\, Centre for Mathematical Sciences
CONTACT:Nathan Bowler
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