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SUMMARY:On model theory\, noncommutative geometry and topoi - Boris Zilber
 \, University of Oxford
DTSTART:20110201T141500Z
DTEND:20110201T151500Z
UID:TALK29695@talks.cam.ac.uk
CONTACT:Nathan Bowler
DESCRIPTION:I will start with a model theoretic notion. Zariski geometries
  is a class of structures discovered to answer a classification problem. T
 he prototypical version of a Zariski geometry is an algebraic variety over
  an algebraically closed field with relations on it given by Zariski close
 d subsets. The initial expectation that Zariski geometries are essentially
  of this kind were proven true to some extent but have generally been over
 turned by new examples (Hrushovski & Zilber\, 1993).\n\nThe present-day in
 terpretation of "new" Zariski geometries leads to Noncommutative geometry.
  For a large class of noncommutative algebras\, e.g. quantum algebras at r
 oots of unity\, we established a duality between the algebras and Zariski 
 geometries as their "co-ordinate algebras"\, typically noncommutative\, ex
 tending the well-known duality between classical geometric objects and the
  algebras of regular (continuous) functions on them.  Zariski geometry in 
 this construction appears essentially as a category of\nrepresentations of
  the algebra. This can be extended to a broader geometric context\, with t
 opology richer than Zariski one.\n\nA different motivation led the physici
 st C.Isham and the philosopher J.Butterfield to suggest a certain kind of 
 topoi as a possible "geometric spaces" for noncommutative "co-ordinate alg
 ebras". This has been\ninvestigated 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
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