**Giovanni Pistone (de Castro Statistics, Collegio Carlo Alberto, Moncalieri)**

It was shown by C. R. Rao in a paper published 1945 that the set of positive probabilities on a finite state space is a *Riemannian manifold*^{1} in a way which is of interest for Statistics because the metric tensor equals the Fisher information matrix. It was later pointed out by Sun-Ichi Amari^{2}, that it is actually possible to define two other affine geometries of Hessian type^{3} on top of the classical Riemannian geometry. Amari gave to this new topic the name of *Information Geometry*.

The term *Algebraic Statistics* was introduced to denote the use of Commutative Computational Algebra in Statistical Design of Experiments and in Statistical Models^{4}. Information Geometry and Algebraic statistics are deeply connected because of the central place occupied by exponential families^{5} in both fields.

The present talk is a tutorial and rigorous introduction the topic. See LINK for some recent research paper.

Manfredo P. do Carmo.

*Riemannian Geometry.*Birkhäuser 1992; Serge Lang.*Differential and Riemannian Manifolds.*Springer 1995.↩The original work of Amari was published in the '80s, see Shun'ichi Amari.

*Differential-geometrical methods in statistics.*Springer 1985. See updated presentations in: Robert E. Kass and Paul W. Vos.*Geometrical Foundations of Asymptotic Inference*. Wiley 1997; Shun'ichi Amari and Hiroshi Nagaoka.*Methods of information geometry*. AMS 2000.↩Hirohiko Shima.

*The Geometry of Hessian Structures.*World Scientiﬁc 2007.↩The term was first used by Giovanni Pistone, Eva Riccomagno, Henry P. Wynn.

*Algebraic Statistics: Computational Commutative Algebra in Statistics.*Chapman & Hall/CRC Monographs on Statistics & Applied Probability 2000. See the Conference Algebraic Statistics 2015, June 8-11, 2015, Department of Mathematics University of Genoa for pointers to current research.↩A classical monograph is by Lawrence D. Brown.

*Fundamentals of statistical exponential families with applications in statistical decision theory.*IMS 1986. Modern algebraic devolopments are in Mateusz Michałek, Bernd Sturmfels, Caroline Uhler, Piotr Zwiernik. "Exponential Varieties." arXiv:1412.6185v1 [math.AG].↩