Title：Stereotype dualities in Geometry
Speaker: Sergei Akbarov (HSE University)
Time: Wednesday, May24, 16:00-17:30
Location：Zoom Meeting, Meeting ID：882 8540 7533, Password：028422
Abstract: The natural parallels between the four big geometric disciplines in mathematics,
- algebraic geometry,
- complex geometry,
- differential geometry,
- topology,
can be explained by the fact that these disciplines appear as a visual image when studying the very same reality with the help of different observation tools.A construction that formalizes this idea is called an envelope. This is a special kind of functor in category theory, generated by a class of morphisms chosen as the observation tools. In the case of the mentioned geometric disciplines, the common reality they reflect is the theory of topological (more precisely, stereotype) algebras, and the functors that transform this reality into the last three disciplines are respectively
- the holomorphic envelope, where the observation tools are the homomorphisms into Banach algebras,
- the smooth envelope, where the observation tools are the so called differential homomorphisms into C*-algebras with the joined self-adjoint nilpotent elements, and
- the continuous envelope, where the observation tools are the homomorphisms into C*-algebras.
Each of these functors generates, apart from the corresponding geometric discipline itself, a special kind of duality in it, which is called stereotype duality, and which generalizes the famous Pontryagin duality for locally compact Abelian groups (to some class of not necessarily commutative groups).
This leads to an intriguing picture, where it becomes possible to compare these geometries as disciplines, to find common features, differences, generalizations, new examples, and so on. In my talk I’m going to give accurate definitions and discuss some details of this picture.
More information about the Functional Analysis Seminar can be found here.
