Event in 188 days!
Submission deadline in 20 days!


As for GSI’13, GSI’15 and GSI’17, the objective of this SEE GSI’19 conference, hosted in Toulouse at ENAC, is to bring together pure/applied mathematicians and engineers, with common interest for Geometric tools and their applications for Information analysis.

It emphasizes an active participation of young researchers to discuss emerging areas of collaborative research on “Geometric Science of Information and their Applications”.

Current and ongoing uses of Information Geometry Manifolds in applied mathematics are the following: Advanced Signal/Image/Video Processing, Complex Data Modeling and Analysis, Information Ranking and Retrieval, Coding, Cognitive Systems, Optimal Control, Statistics on Manifolds, Topology/Machine/Deep Learning, Artificial Intelligence, Speech/sound recognition, natural language treatment, Big Data Analytics, Learning for Robotics, etc., which are substantially relevant for industry.

The Conference will be therefore held in areas of topics of mutual interest with the aim to:

  • Provide an overview on the most recent state-of-the-art

  • Exchange mathematical information/knowledge/expertise in the area

  • Identify research areas/applications for future collaboration

This conference will be an interdisciplinary event and will unify skills from Geometry, Probability and Information Theory. 

GSI’19 will be the opening event of CIMI labex trimester on « Statistics with Geometry and Topology » : https://perso.math.univ-toulouse.fr/statistics-geometry-and-topology/

Proceedings are published in Springer's Lecture Note in Computer Science (LNCS) series.  SPRINGER will sponsor Best paper Award GSI’19.

Gala Diner will take place at Hôtel-Dieu Saint-Jacques in Salle Des Colonnes.


Provisional topics of interests:

  • Probability on Riemannian Manifolds 
  • Optimization on Manifold 
  • Shape Space 
  • Statistics on non-linear data 
  • Lie Group Machine Learning
  • Harmonic Analysis on Lie Groups
  • Statistical Manifold & Hessian Information Geometry 
  • Monotone Embedding in Information Geometry 
  • Non-parametric Information Geometry 
  • Computational Information Geometry 
  • Divergence Geometry 
  • Optimal Transport 
  • Geometric Deep Learning
  • Geometry of Hamiltonian Monte Carlo
  • Information Topology
  • Geometric & (Poly)Symplectic Integrators
  • Geometric structures in thermodynamics and statistical physics
  • Contact Geometry & Hamiltonian Control 
  • Geometric and structure preserving discretizations 
  • Geometry of Quantum States
  • Geodesic Methods with Constraints 
  • Probability Density Estimation & Sampling in High Dimension
  • Geometry of Graphs and Networks
  • Distance Geometry 
  • Geometry of Tensor-Valued Data 
  • Geometric Mechanics
  • Geometric Robotics & Learning
  • Geometry in Neuroscience & Cognitive Sciences 
A special session will deal with:
  • Geometric Science of Information Libraries (geomstats, pyRiemann , …)

Important dates

  • Deadline for 8 pages SPRINGER LNCS format: 18th of February 2019
  • Notification of acceptance: 22nd of April 2019
  • Final paper submission: 15th of June 2019

Provisional program of Invited Speakers:

History Session: J.P. Hiriart-Urruty on “Pierre de Fermat and Blaise Pascal: Geometry & Chance”, F. Barbaresco & M. Boyom on “Tribute to Jean-Louis Koszul”,
Keynote speakers: A. Chenciner on “n-body relative equilibria in higher dimensions”, E. Celledoni on “Structure preserving algorithms for geometric numerical integration”, G. Peyré on “Optimal Transport for Machine Learning”, K. Friston on “Markov blankets and Bayesian mechanics”.


The mathematician Élie CARTAN (1869-1951) was born in Dolomieu. Son of a blacksmith, this young boy is noticed by the teacher of the communal school of Dolomieu. He appointed him to the attention of the cantonal delegate, Antonin DUBOST, who later became President of the Senate. A. DUBOST takes under his protection the young Elie and makes him study at the College of Vienne, then at the Lycee of Grenoble and finally at the Lycee Janson-de-Sailly in Paris. There, after a year of preparation, Élie CARTAN was received at the École Normale Supérieure in 1888. His doctoral thesis (1894), devoted to the classification of Lie groups, was a historic event. His later work on complex semi-simple groups, then real, had him appointed at the Sorbonne in 1909.
What made him famous all over the world was on the one hand his collaboration, after 1920, with the German mathematician Hermann WEYL, on the global study of Lie groups, on the other hand, the use he did the theory of groups of differential Geometry, then his theory of "generalized spaces" which was applicable in the theory of Relativity and which was the occasion of a long exchange of letters between Albert EINSTEIN and Elie CARTAN. These letters were published by Princeton University Press in 1979, on the occasion of the centenary of the birth of EINSTEIN. It is also to Elie CARTAN that we owe the systematic introduction of "differential exterior forms" in Differential Geometry.
Elie CARTAN was elected to the Academy of Sciences "in 1931. In 1969, the centenary of his birth was celebrated by international symposia in Bucharest and Grenoble, a street of which bears his name.
Elie Cartan papers at archive of the French Academy of Science:
List of documents in Elie Cartan archive:
biographical note:

A reserved, kindly man and a diligent worker, Eugène Cosserat was one of the moving forces in the University of Toulouse for thirty five years. He studied the deformation of surfaces which led him to a “theory of elasticity”. The Cosserat brothers, following a suggestion by Duhem (1893), developed a theory for continuous oriented bodies that consist not just of particles (or material points), but also of directions associated with each particle. Eugène Cosserat died in his home at the Observatory in Toulouse.

See Cosserat biography by J J O'Connor and E F Robertson


At the age of 17 he took the competitive entrance examinations for the two major Paris Institutions, the École Polytechnique and the École Normale Supérieure, and was offered a place at both. Unlike his two brothers who both studied at the École Polytechnique, he chose to study at the École Normale Supérieure which he entered in 1883. During three years of study at the École Normale, Cosserat attended lectures by leading mathematicians including Paul Appell, Gaston Darboux, Gabriel Koenigs and Émile Picard. Among his fellow students were several who would make major contributions to mathematics, including Jacques Hadamard and Paul Painlevé. Cosserat graduated in 1886 and spent a short time teaching at the Lycée in Rennes before he was appointed as an assistant astronomer at the Observatory in Toulouse towards the end of 1886.

Even before the award of his doctorate in 1889, Cosserat had begun teaching mathematics courses at the Faculty of Science at Toulouse. In 1896 he became professor of differential and integral calculus there, replacing Thomas Stieltjes who had died on 31 December 1894, and, from that time on, he divided his work between the Faculty of Science and the Observatory. In 1908 Cosserat was appointed to the chair of astronomy at Toulouse, becoming director of the Observatory there for the rest of his life. In this latter  role he replaced Édouard Benjamin Baillaud who had left Toulouse to become director of the Paris Observatory. The role of director of the Observatory was a demanding one, and Cosserat became almost totally occupied with administrative tasks from the time of his appointment and so was forced to essentially give up mathematical research from this time on.

Although he was not living in Paris, Cosserat was elected to the Académie des Sciences as a corresponding member on 19 June 1911 and a full member on 31 March 1919. Four years later, he was elected to the Bureau de Longitude. Because he was in Toulouse rather than Paris, he was made a non-resident member of both these organisations. In 1889 he was awarded the Poncelet Prize by the Académie des Sciences.

In mathematics, we have already noted his early work on geometry. In his later work, Cosserat studied the deformation of surfaces which led him to a theory of elasticity. This work was carried out in collaboration with his brother, François Cosserat, who was an engineer. He began his collaboration with his brother in 1896 with the publication Théorie de l'élasticité. This first work studied broad questions relating to the foundations of mechanics but later their work turned towards the physical theory. By the early 1900s, Cosserat had stopped working on the type of geometrical problems that had interested him at the start of his career and all his research efforts were directed towards working on mechanics with his brother. Their most important joint publications are: Note sur la cinématique d'un milieu continu (1897); Note sur la dynamique du point et du corps invariable  (1906); Note sur la théorie de l'action euclidienne  (1909); and the book Théorie des corps déformables  (1909). The first of these was published as an addition to Gabriel Koenigs Leçons de Cinématique professées à la Sorbonne: cinématique théorique. A review of this work by E O Lovett in the Bulletin of the American Mathematical Society in 1900 singles out the Cosserats' contribution:-

The introduction of this note is peculiarly fortunate for it is high time that kinematics should comprehend the study of deformation and of deformable spaces. The authors have included in their extract certain generalities on curvilinear coordinates, the deformation of a continuous medium in general, infinitely small deformation, use of the mobile trieder, and the case where the non-deformed medium is referred to any curvilinear coordinates.

This innovative work on mechanics (21 joint publications on this topic are listed in [2]) ended with the François Cosserat's death in 1914, after which time his brother Eugène Cosserat published nothing further on the topic. Jacques Levy describes the two Cosserats' contributions to this area [1]:-

The most practical results concerning elasticity were the introduction of the systematic use of the movable trihedral and the proposal and resolution, before Fredholm's studies, of the functional equations of the sphere and ellipsoid. Cosserat's theoretical research, designed to include everything in theoretical physics that is directly subject to the laws of mechanics, was founded on the notion of Euclidean action [least action] combined with Lagrange's ideas on the principle of extremality and Lie's ideas on invariance in regard to displacement groups. The bearing of this original and coherent conception was diminished in importance because at the time it was proposed, fundamental ideas were already being called into question by both the theory of relativity and progress in physical theory.

The authors of [5] write:-

The Cosserat brothers, following a suggestion by Duhem (1893), developed a theory for continuous oriented bodies that consist not just of particles (or material points), but also of directions associated with each particle. Thus, in addition to the field of position vectors of a continuum in a given configuration, one also admits vector fields ... which may be chosen so as to represent pertinent features of materials. ... The Cosserats themselves recognised the value of oriented two-dimensional continua (i.e., curves and surfaces endowed with additional structure in the form of directors) for representing the deformations of rods and shells respectively. ... [However their] ideas on the subject [were] ignored for half a century.

Another aspect of Eugène Cosserat's work which we should mention is his contributions to the Annales de la faculté des sciences de Toulouse. This journal began publication in 1887 and, two years later, Cosserat joined the editorial board. The two other mathematicians who served on this board at this time were Henri Andoyer and Thomas Jan Stieltjes. In 1896 Cosserat became secretary to the editorial board of the Annals and he continued to hold this role until 1930. In fact, he continued to undertake editorial work up to the time of his death, sending Henri Poincaré a letter on an editorial matter just a few days before his death.

Eugène Cosserat died in his home at the Observatory in Toulouse [4]:-

The funeral took place on 2 June, on a morning with gentle sun; a long procession descended from the Observatory along the slopes which, although close to the city, still retained some greenery. It seemed that Nature had staged a scene both bright and calm ... calm as he was in his way.

  1. J R Levy, Biography in Dictionary of Scientific Biography (New York 1970-1990).


  1. J F Pommaret, Lie pseudogroups and mechanics (Taylor & Francis, 1988).


  1. M Brocato and K Chatzis, Les Frères Cosserat. Brève Introduction à Leur Vie et à Leurs Travaux en Mécanique.

  2. A Buhl, Eugène Cosserat. Annales de la faculté des sciences de Toulouse 23 (1931), v-viii.

  3. J Casey and M J Crochet, Paul M Naghdi (1924-1994) in J Casey and M J Crochet (eds.), Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids: a collection of papers in honor of Paul M Naghdi (Birkhäuser, 1995), S1-S32.

  4. P Caubet, E Cosserat: set vues générales sur I'astronomie de position, Journal des observateurs, 14 (1931), 139-143

  5. L Montangerand, Eloge de E Cosserat lu à la séance du 30 juin 1932 de l'Académie des Sciences, Inscriptions et Belles-Lettres de Toulouse, Ann. de l'Observatoire de Toulouse 10 (1933), xx-xxx.

  6. Cosserat, E.; Cosserat, F. (1909). Théorie des Corps deformables. Paris: A, Hermann et Fils

French mathematician - https://www.youtube.com/watch?v=2UC7TWCJWkM

Written By: Carl B. Boyer

Pierre de Fermat, (born August 17, 1601, Beaumont-de-Lomagne, France—died January 12, 1665, Castres), French mathematician who is often called the founder of the modern theory of numbers. Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. Independently of Descartes, Fermat discovered the fundamental principle of analytic geometry. His methods for finding tangents to curves and their maximum and minimum points led him to be regarded as the inventor of the differential calculus. Through his correspondence with Blaise Pascal he was a co-founder of the theory of probability.

Life and early work

Little is known of Fermat’s early life and education. He was of Basque origin and received his primary education in a local Franciscan school. He studied law, probably at Toulouse and perhaps also at Bordeaux. Having developed tastes for foreign languages, classical literature, and ancient science and mathematics, Fermat followed the custom of his day in composing conjectural “restorations” of lost works of antiquity. By 1629 he had begun a reconstruction of the long-lost Plane Loci of Apollonius, the Greek geometer of the 3rd century bce. He soon found that the study of loci, or sets of points with certain characteristics, could be facilitated by the application of algebra to geometry through a coordinate system. Meanwhile, Descartes had observed the same basic principle of analytic geometry, that equations in two variable quantities define plane curves. Because Fermat’s Introduction to Loci was published posthumously in 1679, the exploitation of their discovery, initiated in Descartes’s Géométrie of 1637, has since been known as Cartesian geometry. In 1631 Fermat received the baccalaureate in law from the University of Orléans. He served in the local parliament at Toulouse, becoming councillor in 1634. Sometime before 1638 he became known as Pierre de Fermat, though the authority for this designation is uncertain. In 1638 he was named to the Criminal Court

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 https://www.youtube.com/watch?v=AzK5K7Q05sw

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 ://www.youtube.com/watch ?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 https://www.springer.com/la/book/9789811339868
[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 ://www.mccme.ru/tg2017/slides/alexeevsky.pdf  ,vidéo : http ://www.mathnet.ru/present19121
[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;  https://link.springer.com/chapter/10.1007/978-3-030-02520-5_12
[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.;  https://www.springer.com/la/book/9789811339868
[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 ; http://smai.emath.fr/IMG/pdf/Matapli116.pdf (long version: http://forum.cs-dc.org/uploads/files/1520499744825-jean-louis-koszul-et-les-structures-%C3%A9l%C3%A9mentaires-de-la-g%C3%A9om%C3%A9trie-de-l-x27-information-final-revc-1.pdf )
[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 ; http://gretsi.fr/colloque2017/myGretsi/programme.php ;
[29] FGSI’19 Cartan-Koszul-Souriau « Foundations of Geometric Structure of Information », 4-6 Février 2019, IMAG Montpellier ; https://fgsi2019.sciencesconf.org/


Comité d'organisation

  • Fr Barbaresco - THALES LAS FRANCE SAS http://www.thalesgroup.com
  • Ludovic D'estampes - ENAC
  • Thierry Klein - ENAC
  • Alice Le brigant - Université de Bordeaux
  • Florence Nicol - Ecole Nationale de l'Aviation Civile
  • Frank Nielsen - Ecole Polytechnique, France http://www.lix.polytechnique.fr/~nielsen/
  • Stephane Puechmorel - ENAC
  • Tat-Dat To - ENAC

Sponsors and Organizers


ENAC, Toulouse (France)

7, avenue Edouard BelinCS 54005
31055 Toulouse Cedex 4
GPS ccordinates GPS : 43.565156, 1.479281
Ecole Nationale de l'Aviation Civile
7, avenue Edouard Belin CS 54005
31055 Toulouse Cedex 4


Shuttle from Toulouse-Blagnac Airport

Free "Airport" shuttle, reserved for ENAC students, teachers and speakers according to availability.
Boarding on presentation of notification or student / trainee card.

By public transport

All information, maps and directions for public transportation in Toulouse are available on Tisseo website ///

  • N°68 - to La terrasse / Métro Ramonville
  • N°78 - to Université Paul Sabatier / Lycée St Orens
  • N°37 - to Jolimont / Métro Ramonville

Subway - Line B

  • Get off at Faculté de pharmacieFaculty and take bus N°78 to Lycée St Orens
  • Get off at Ramonville-Saint-Agne and take bus N°68 to "La terrasse"

Subway - Line  A

Get off at Jolimont and take bus N°37 to Ramonville Metro or bus N°68

By car
Take outer ring road (towards "Montpellier"), then follow "Toulouse center / Foix / Tarbes" (green sign). Exit N°20, follow "Complexe scientifique Rangueil".
GPS : 43.565156, 1.479281