Abstract
IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.
Highlights
(1) The introduction of a new theory of statistical models. This re-establishment must answer both questions of McCullagh and Gromov; (2) The search for an characteristic invariant which encodes the points of the moduli space of isomorphism class of models; (3) The introduction of the theory of homological statistical models
We address its links with Hessian geometry; (4) We emphasize the links between the classical theory of models, the new theory and Vanishing Theorems in the theory of homological statistical models
We introduce the theory of homological statistical model and we explore the links between this theory and the challenge 2
Summary
Throughout the paper we use tha following notation. N is the set of non negative integers, Z is the ring of integers, R is the field of real numbers, C∞(M) is the associative commutative algebra of real valued smooth functions in a smooth manifold M. Let ∇ be a Koszul connection in a manifold M, R∇ is the curvature tensor of ∇. To a pair of Koszul connections (∇, ∇∗) we assign three differential operators. They are denoted by D∇∇∗ , D∇ and D∇. (A.1) D∇∇∗ is a first order differential operator. It is defined in the vector bundle Hom(TM, TM). Its values belong to the vector bundle Hom(TM⊗2, TM). (A.2) D∇ and D∇ are 2nd order differential operators They are defined in the vector bundle TM. In the Appendix A to this paper we overview the role played by the solutions to FE∗∗(∇) in some still open problems
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