Vinberg's theory of homogeneous convex cones: developments and applications

The talk is a review of developments and applications of selected basic ideas, results and notions, presented by E.B. Vinberg in his papers about classification of homogeneous convex cones. We confine ourselves to only three topics:

• Theory of left symmetric algebras (Vinberg–Koszul algebras) and Hessian manifolds.
• Information geometry , i.e. the geometry of manifolds of probability measures , developed mostly by N.N. Chentsov and S-I Amary and based on ideas by R.A.Fisher, C.R.Rao, C. Shannon and S. Kullback.
• Supergravity. Application of the theory of matrix $T$-algebras by Vinberg for description of so called special geometries (very special real geometry, special Kähler geometry and special quaternionic Kähler geometry (affine and projective)) which arise as matter multiplets in Supersymmetry and Supergravity in spacetime dimension $d = 6,5,4,3$.