Tribute to Jean-Louis Koszul (1921 – 2018)

Jean-Louis Koszul passed away January 12th 2018. This tribute is a scientific exegesis and admiration of Jean-Louis Koszul’s works on homogeneous bounded domains that have appeared over time as elementary structures of Information Geometry. Koszul has introduced fundamental tools to characterize the geometry of sharp convex cones, as Koszul-Vinberg characteristic Function, Koszul Forms, and affine representation of Lie Algebra and Lie Group. The 2nd Koszul form is an extension of classical Fisher metric. Koszul theory of hessian structures and Koszul forms could be considered as main foundation and pillars of Information Geometry.

The community of “Geometric Science of Information” (GSI) has lost a mathematician of great value, who informed his views by the depth of his knowledge of the elementary structures of hessian geometry and bounded homogeneous domains. His modesty was inversely proportional to his talent. Professor Koszul built in over 60 years of mathematical career, in the silence of his passions, an immense work, which makes him one of the great mathematicians of the XX’s century, whose importance will only affirm with the time. In this troubled time and rapid transformation of society and science, the example of Professor Koszul must be regarded as a model for future generations, to avoid them the trap of fleeting glories and recognitions too fast acquired. The work of Professor Koszul is also a proof of fidelity to his masters and in the first place to Prof. Elie Cartan, who inspired him throughout his life. Henri Cartan writes on this subject “I do not forget the homage he paid to Elie Cartan’s work in Differential Geometry during the celebration, in Bucharest, in 1969, of the centenary of his birth. It is not a coincidence that this centenary was also celebrated in Grenoble the same year. As always, Koszul spoke with the discretion and tact that we know him, and that we love so much at home”.

We will conclude by quoting Jorge Luis Borges “both forgetfulness and memory are apt to be inventive” (Doctor Brodie’s report). Our generation and previous one have forgotten or misunderstood the depth of the work of Jean-Louis Koszul and Elie Cartan on the study of bounded homogeneous domains. It is our responsibility to correct this omission, and to make it the new inspiration for the Geometric Science of Information. We will invite readers  to listen to the last interview of Jean-Louis Koszul for 50th birthday of Joseph Fourier Institute

Frédéric Barbaresco & Michel Boyom


[1] Selected Papers of J L Koszul, Series in Pure Mathematics, Volume 17, World Scientific Publishing, 1994
[2] Cartan H., Allocution de Monsieur Henri Cartan, colloque Jean-Louis Koszul, Annales de l’Institut Fourier, tome 37, 4, pp.1-4, 1987
[3] Koszul J.L., L’oeuvre d’élie Cartan en géométrie différentielle, in élie Cartan, 1869-1951. Hommage de l’Académie de la République Socialiste de Roumanie à l’occasion du centenaire de sa naissance. Comprenant les communications faites aux séances du 4e Congrès du Groupement des Mathématiciens d’Expression Latine, tenu à Bucarest en 1969 (Editura Academiei Republicii Socialiste Romania, Bucharest, 1975), pp. 39-45.
[4] dernière interview de J.L. Koszul pour le laboratoire mathématique de l’Institut Fourier en 2016 : vidéo : https :// ?v=AzK5K7Q05sw
[5] Koszul J.L. , Sur la forme hermitienne canonique des espaces homogènes complexes. Can. J. Math., 7, 562–576, 1955
[6] Koszul J.L. , Exposés sur les Espaces Homogènes Symétriques ; Publicação da Sociedade de Matematica de São Paulo : São Paulo, Brazil, 1959
[7] Koszul J.L., Domaines bornées homogènes et orbites de groupes de transformations affines, Bull. Soc. Math. France 89, pp. 515-533., 1961
[8] Koszul J.L. , Ouverts convexes homogènes des espaces affines. Math. Z., 79, 254–259, 1962
[9] Koszul J.L. , Variétés localement plates et convexité. Osaka. J. Math., 2, 285–290, 1965
[10] Koszul J.L, Lectures on Groups of Transformations, Tata Institute of Fundamental Research, Bombay, 1965
[11] Koszul J.L., Déformations des variétés localement plates, .Ann Inst Fourier, 18 , 103-114, 1968
[12] Koszul J.L., Trajectoires Convexes de Groupes Affines Unimodulaires. In Essays on Topology and Related Topics ; Springer : Berlin, Germany, pp. 105–110, 1970
[13] Koszul J.L., ZOU Y. Introduction to Symplectic Geometry Springer: Berlin, Germany, 2019
[14] Vey J., Sur une notion d’hyperbolicité des variétés localement plates, Thèse de troisième cycle de mathématiques pures, Faculté des sciences de l’université de Grenoble, 1969
[15] Vey J., Sur les automorphismes affines des ouverts convexes saillants, Annali della Scuola Normale Superiore di Pisa, Classe di Science, 3e série, tome 24, 4, p.641-665, 1970
[16] Alekseevsky D., Vinberg’s theory of homogeneous convex cones : developments and applications, Transformation groups 2017. Conference dedicated to Prof. Ernest B. Vinberg on the occasion of his 80th birthday, Moscou, December, 2017, https ://  ,vidéo : http ://
[17] Vinberg E.B., Homogeneous cones, Dokl. Akad. Nauk SSSR., 133, pp. 9–12, 1960 ; Soviet Math. Dokl., 1, pp. 787–790, 1961
[18] Vinberg E.B., The structure of the group of automorphisms of a convex cone, Trudy Moscov. Mat. Obshch., 13, pp.56–83, 1964 ; Trans. Moscow Math. Soc., 13, 1964
[19] Shima H., The Geometry of Hessian Structures, World Scientific, 2007
[20] Shima H., Geometry of Hessian Structures, Springer Lecture Notes in Computer Science, Vol. 8085, F. Nielsen, & Barbaresco, Frederic (Eds.), pp.37-55, 2013
[21] Malgrange B., Quelques souvenirs de Jean-Louis KOSZUL, Gazette des Mathématiciens - 156, pp. 63-64, Avril 2018
[22] Cartier P., In memoriam Jean-Louis KOSZUL, Gazette des Mathématiciens - 156, pp. 64-66, Avril 2018
[23] Nguiffo Boyom M., Transversally Hessian foliations and information geometry I. Am. Inst. Phys. Proc. , 1641, pp. 82–89, 2014
[24] Nguiffo Boyom M, Wolak, R., Transverse Hessian metrics information geometry MaxEnt 2014. AIP. Conf. Proc. Am. Inst. Phys. 2015
[25] Barbaresco, F. Jean-Louis Koszul and the elementary structures of Information Geometry. In Geometric Structures of Information Geometry; Nielsen, F.; Ed.; Springer: Berlin, Germany, 2018;
[26] Barbaresco, F. Koszul Contemporaneous Lectures: Elementary Structures of Information Geometry and Geometric Heat Theory. In Introduction to Symplectic Geometry; Koszul, J.L., Ed.; Springer: Berlin, Germany, 2018.;
[27] Barbaresco, F. Jean-Louis Koszul et les Structures Elémentaires de la Géométrie de l’Information; Revue SMAI Matapli; SMAI Editor; Volume 116, pp.71-84, Novembre 2018 ; (long version: )
[28] Barbaresco, F. Les densités de probabilité « distinguées » et l'équation d'Alexis Clairaut: regards croisés de Maurice Fréchet et de Jean-Louis Koszul, Conférence Histoire de la discipline, GRETSI'17 , Juan-Les-Pins, Septembre 2017 ; ;
[29] FGSI’19 Cartan-Koszul-Souriau « Foundations of Geometric Structure of Information », 4-6 Février 2019, IMAG Montpellier ;