Cohomology in algebra, geometry, physics and statistics

The purpose of this talk is to present a connection between the mathematical entities mentioned in the title. It will be argued that Jordan algebras provide a suitable playground in which parametric models of classical and quantum information geometry can joyfully play (and hopefully thrive). In order to recover the Riemannian geometry of parametric models extensively used in classical and quantum information geometry, the method of coadjoint orbits will be adapted to Jordan algebras. Indeed, given the symmetric nature of the Jordan product, the analogue of the Konstant-Kirillov-Souriau symplectic form becomes a symmetric covariant tensor field. When suitable choices of Jordan algebras are made, it is possible to recover the Fisher-Rao metric tensor characteristic of classical information geometry or the Bures-Helstrom metric tensor appearing in quantum information geometry. This instance tells us that geometrical structures in information geometry can be found looking at... more