The last formula of Jean-Louis Koszul
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The last formula of Jean-Louis Koszul Michel Nguiffo Boyom1 Received: 6 April 2020 / Revised: 1 September 2020 / Accepted: 11 September 2020 © Springer Nature Singapore Pte Ltd. 2020
Abstract Since the first international conference in France on the Information Geometry, GSI2013, Jean-Louis Koszul’s interest in Information Geometry went increasing. Motivated by the impact of the cohomology of Koszul-Vinberg algebras on the Information Geometry and moved by some issues raised by Albert Nijenhuis, Jean-Louis Koszul undertook another rewriting of the Brut formula of the coboundary operator of the KV complex. The source of other motivations of Jean-Louis Koszul was the relationships between the theory of KV cohomology and the theory of deformation of locally flat manifolds.In 2015 Jean-Louis Koszul sent me his Last formula of the KV boundary operator. In that Last formula Jean-Louis Koszul dealt with the case where spaces of coefficients are trivial modules of KV algebras. A part of the present work is devoted to extending the Last formula of Jean-Louis Koszul to KV cochain complexes whose spaces of coefficients are non-trivial two-sided modules of KV algebras. At another side, I also aim to highlight other significant impacts of the theory of KV cohomology of Koszul-Vinberg algebras. In particular I will use the KV cohomology to widely revisit the theory of statistical models of measurable sets. The reader will see why the source of the theory of statistical models is of homological nature.I also intend to highlight several impacts of the KV cohomology on the quantitative differential topology. I am particularly concerned with problems regarding the existence of Riemannian foliations, the existence of symplectic foliations as well as the existence of multi-dimensional webs. The homological theory of statistical models is presented as branches of rooted trees whose roots are weakly Jensen random cohomology classes. Keywords Lie algebroids · KV cohomology · Canonical characteristic class · Koszul geometry · Functor of Amari · Locally flat manifolds · Complex systems Mathematics Subject Classification Primaries 53B05 · 53C12 · 53C16 · 22F50 · Secondarie 18G60 TRIBUTE TO JEAN-LOUIS KOSZUL 03 January 1921–12 January 2018.
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Michel Nguiffo Boyom [email protected] IMAG:Alexander Grothendieck Research Institute, UMR CNRS 5149, University of Montpellier, Montpellier, France
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Information Geometry
1 Introduction Je suis fort content d’apprendre que vous avez invité Shima à votre congrès GSI. Il a été je crois l’un des premiers, sinon le premier, à voir que la géométrie Hessienne avait des affinités inattendues. J-L Koszul 12 Décembre 2012 I am truly delighted that you have invited Shima to your congress GSI. He was one of the first, if not the first, to see that the Hessian Geometry has unexpected affinities. J-L Koszul 12 December 2012 The first international conference in France on the Information Geometry took place in Ecole des Mines de Paris in October 2013. One year before both the organizing committee and
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