Geometry of noncommutative spaces
Geometry of noncommutative spaces - Dr. Asghar Ghorbanpour (Western)
Date / Time: Friday, April 6th @ 3:30 P.M.
Location: ARTS 108
Speaker: Dr. Asghar Ghorbanpour (Western)
Title: Geometry of noncommutative spaces
Abstract: The discovery of the relation between the category of Hausdorff compact topological spaces and the category of commutative unital C*-algebras (the Gelfand-Naimark theorem) in the mid-twentieth century, as well as the evolution of topological theories for noncommutative C*-algebras such as K-theory, established a theory that can be called "noncommutative topology". The notion of geometry for these noncommutative spaces, however, was not introduced until later, when Alain Connes showed that the geometry can be formulated using spectral triples in which the geometric information is basically encoded in the spectrum of an operator D, the Dirac operator, which has compact resolvent and its commutator with the elements of the algebra is bounded.
The main focus of this talk will be on how the spectrum of D can be used to recover some geometric notions and quantities such as volume and dimension. In particular, we shall discuss recent developments in introducing the scalar curvature and Ricci functional.