International Workshop on
Affine Differential Geometry
and Information Geometry
Jun Zhang (
(Nagoya Institute of
(Please contact: email@example.com for more information)
Sanae Kurosu (Tokyo Institute of Technology)
An-Min Li (Sichuan University)
(Nagoya Institute of
Huafei Sun (Beijing Institute of Technology)
Jun Zhang (
TITLE AND ABSTRACT
Gauge Invariance in Affine Hypersurface Theory: Some New Developments (UDO SIMON)
The Complete Affine Hypersurfaces with Negative Constant Affine Mean Curvature (BAOFU WANG, jointly with AN-MIN LI)
Given a bounded convex domain D in n-dimensional Eulidean space and a boundary value d we construct a Euclidean complete hypersurface with constant affine mean curvature L < 0.
The Second Variation Formula for Minimal Centroaffine Immersion of Codimension Two (HUAFEI SUN, jointly with MITSUKI OURA and KOTARO YAMADA)
The second variation formula of centroaffine minimal surfaces are derived. In contrast to the case of codimension one, for which the second variation of the area for any variation of affine surfaces with definite affine fundamental form are negative, several examples of centroaffine surface of codimension two with definite affine fundamental form, which admit variations with positive second
variation of the area functional are given.
Basic Correspondences Between Affine Differential Geometry and Information Geometry (HIROSHI MATSUZOE)
Affine differential geometry studies hypersurfaces immersed into an affine space, and information geometry studies geometrical structures of sets of probability distributions. A key fact is that duality of affine connections arises naturally, so these geometries have common geometric ideas. In this talk, a tutorial of correspondences between affine differential geometry and information geometry will be given.
We construct a class of divergence functions based on convex analysis, and then show that they induce the structure of statistical manifold with alpha-connections. We then discuss representations by biorthogonal coordinates, which arise out of statistical context as natural and expectation parameters of statistical models.
We investigate the geometry of the space of positive definite matrices induced by the class of convex functions called V-potentials from the viewpoints of information and affine geometry. We show the geometry is invariant for the unimodular group action and naturally induces a foliated structure. Each leaf is proved to be a statistical manifold with a negative constant curvature and possesses the special decomposition property of a pseudo-distance function called the divergence. The result can be extended to general symmetric cones.
In the usual (or classical) information geometry, a manifold of probability distributions is provided with a Riemannian metric called the Fisher metric and a one-parameter family of affine connections called the alpha-connections. In particular, the triplet of the Fisher metric, the 1-connection and the (-1)-connection can be identified with the geometric structure induced from the relative entropy (Kullback-Leibler information divergence)through the theory of dual (conjugate) affine connections, and gives a powerful theoretical framework to study many mathematical problems in statistics, information theory and probability theory. In my talk, I would like to give a quick review on recent attempts to extend these notions of information geometry to the quantum mechanical setting where a manifold of quantum states (density operators) is to be treated, focusing upon the similarity and difference to the classical case, mathematical difficulties originating from noncommutativity of operators, and relations to quantum extension of statistical inference and information theory.