Mathematical aspects of information geometry

In this talk, I shall explore the connections of information geometry with various mathematical fields. The Fisher metric is seen as the natural metric on an infinite dimensional projective space. This also yields a geometric interpretation of Green functions (propagators) of quantum field theory. In finite dimensional situations, the Fisher metric induces a pair of dual affine structures. Such a geometry is called K\"ahler affine or Hessian. I shall define a natural differential operator associated to such a structure, the affine Laplacian, and discuss an existence theorem for affine harmonic mappings.