José E. Galé

Some differential geometry in operator theory

We discuss differential geometric properties of several important sets of elements which play an important role in
functional analysis and operator theory.
These sets include orbits of representations,
Grassmann manifolds of subspaces in Banach or Hilbert spaces,
completely positive mappings, and reproducing kernels.
This wide collection of structures can be studied in
a unifying manner in a framework involving category theory.
We shall also point out potential implications of
this geometric point of view for the theoretical physics.
The talk is partly based on my joint work with D. Beltita.