Abstract

I will start with a model theoretic notion. Zariski geometries is a class of structures discovered to answer a classification problem. The prototypical version of a Zariski geometry is an algebraic variety over an algebraically closed field with relations on it given by Zariski closed subsets. The initial expectation that Zariski geometries are essentially of this kind were proven true to some extent but have generally been overturned by new examples (Hrushovski & Zilber, 1993).

The present-day interpretation of “new” Zariski geometries leads to Noncommutative geometry. For a large class of noncommutative algebras, e.g. quantum algebras at roots of unity, we established a duality between the algebras and Zariski geometries as their “co-ordinate algebras”, typically noncommutative, extending 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 representations of the algebra. This can be extended to a broader geometric context, with topology richer than Zariski one.

A different motivation led the physicist C.Isham and the philosopher J.Butterfield to suggest a certain kind of topoi as a possible “geometric spaces” for noncommutative “co-ordinate algebras”. This has been investigated in depth by A.Doering (Oxford). As it turned out the two approaches have a lot in common.

I will report on a recent progress in understanding these connections.