{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T15:04:09Z","timestamp":1775142249158,"version":"3.50.1"},"reference-count":65,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2016,12,2]],"date-time":"2016-12-02T00:00:00Z","timestamp":1480636800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.<\/jats:p>","DOI":"10.3390\/e18120433","type":"journal-article","created":{"date-parts":[[2016,12,2]],"date-time":"2016-12-02T10:36:37Z","timestamp":1480674997000},"page":"433","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology"],"prefix":"10.3390","volume":"18","author":[{"given":"Michel","family":"Boyom","sequence":"first","affiliation":[{"name":"ALEXANDER GROTHENDIECK INSTITUTE, IMAG-UMR CNRS 5149-c.c.051, University of Montpellier, PL. E. Bataillon, F-34095 Montpellier, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,12,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Faraut, J., and Koranyi, A. (1994). Analysis on Symmetric Cones, Oxford University Press.","DOI":"10.1093\/oso\/9780198534778.001.0001"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"103","DOI":"10.5802\/aif.279","article-title":"D\u00e9formation des connexions localement plates","volume":"18","author":"Koszul","year":"1968","journal-title":"Ann. Inst. Fourier"},{"key":"ref_3","first-page":"303","article-title":"The theory of homogeneous convex cones","volume":"12","author":"Vinberg","year":"1963","journal-title":"Trans. Moscow. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Barbaresco, F. (2016). Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Geometry: Applications in Information Geometry. Entropy.","DOI":"10.20944\/preprints201608.0078.v2"},{"key":"ref_5","unstructured":"Nguiffo Boyom, M., and Wolak, R. (2013). Geometric Science of Information, Springer."},{"key":"ref_6","unstructured":"Gindikkin, S.G., Pjateckii, I.I., and Vinerg, E.B. (1968). Geometry of Homogeneous Bounded Domains, Cremonese."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"303","DOI":"10.5802\/aif.302","article-title":"Hyperbolische komplexe Rume","volume":"18","author":"Kaup","year":"1968","journal-title":"Ann. Inst. Fourier"},{"key":"ref_8","first-page":"285","article-title":"Vari\u00e9t\u00e9 localement plate et convexit\u00e9","volume":"2","author":"Koszul","year":"1965","journal-title":"Osaka J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"119","DOI":"10.2140\/pjm.2006.225.119","article-title":"The Cohomology of Koszul-Vinberg Algebras","volume":"225","year":"2006","journal-title":"Pac. J. Math."},{"key":"ref_10","first-page":"237","article-title":"R\u00e9ductions Kahl\u00e9riennes dans les groupes de Lie R\u00e9solubles et Applications","volume":"47","year":"2010","journal-title":"Osaka J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"91","DOI":"10.5802\/aif.794","article-title":"Homogeneous hessian manifolds","volume":"30","author":"Shima","year":"1980","journal-title":"Ann. Inst. Fourier"},{"key":"ref_12","first-page":"622","article-title":"Une notion d\u2019hyperbolicit\u00e9 sur les vari\u00e9t\u00e9s localement plates","volume":"266","author":"Vey","year":"1968","journal-title":"C. R. Acad. Sci. Paris"},{"key":"ref_13","unstructured":"Barbaresco, F. (2013). Matrix Information Geometry, Springer."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1063\/1.4905981","article-title":"Topology forms of informations","volume":"1641","author":"Baudot","year":"2014","journal-title":"AIP Conf. Proc."},{"key":"ref_15","unstructured":"Gromov, M. In a Search for a Structure, Part I: On Entropy. Available online: www.ihes.fr\/gromov\/PDF\/structure-serch-entropy-july5-2012.pdf."},{"key":"ref_16","unstructured":"Gromov, M. (2014, January 21\u201326). On the structure of entropy. Proceedings of the 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering MaxEnt 2014, Amboise, France."},{"key":"ref_17","unstructured":"Amari, S.-I. (1990). Differential Geometry Methods in Statistics, Springer. Lecture Notes in Statistics."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Amari, S.-I., and Nagaoka, H. (2007). Methods of Information Geometry, Translations of Mathemaical Monographs, American Mathematical Society.","DOI":"10.1090\/mmono\/191"},{"key":"ref_19","unstructured":"Arnaudon, M., Barbaresco, F., and Yan, L. (2013). Matrix Information Geometry, Springer."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1112\/S1461157010000513","article-title":"Meadians and means in Fisher geometry","volume":"15","author":"Armaudon","year":"2012","journal-title":"LMS J. Comput. Math."},{"key":"ref_21","first-page":"335","article-title":"Differential geometry and statistics: Some mathematical aspects","volume":"29","year":"1987","journal-title":"Indian J. Math."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Murray, M.K., and Rice, J.W. (1993). Differential Geometry and Statistics, Chapman and Hall\/CRC.","DOI":"10.1007\/978-1-4899-3306-5"},{"key":"ref_23","unstructured":"Nguiffo Boyom, M., and Byande, P.M. (2013). Matrix Information Geometry, Springer."},{"key":"ref_24","unstructured":"Barndorff-Nielsen, O.E. (1978). Information and Exponential Families in Stattistical Theory, Wiley."},{"key":"ref_25","unstructured":"Nguiffo Boyom, M., Jamali, M., and Shahid, M.H. (2015). Geometric Science of Information, Springer."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/0001-8708(77)90004-4","article-title":"The fundamental groups of complete affinely flat manifolds","volume":"25","author":"Milnor","year":"1977","journal-title":"Adv. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"59","DOI":"10.2307\/1970484","article-title":"On deformations of Rings and Algebras","volume":"79","author":"Gerstenhaber","year":"1964","journal-title":"Ann. Math."},{"key":"ref_28","first-page":"225","article-title":"Sur une classe de propri\u00e9t\u00e9s communes \u00e0 quelques types diff\u00e9rents d\u2019alg\u00e8bres","volume":"14","author":"Nijenhuis","year":"1968","journal-title":"Enseign. Math."},{"key":"ref_29","first-page":"197","article-title":"KV Cohomology and diffrential geometry of locally flat manifolds. Information geometry","volume":"14","author":"Byande","year":"2012","journal-title":"Afr. Diaspora J. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1225","DOI":"10.1214\/aos\/1035844977","article-title":"What is statistical model?","volume":"30","author":"McCullagh","year":"2002","journal-title":"Ann. Stat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3253","DOI":"10.3390\/e17053253","article-title":"The homological nature of Entropy","volume":"17","author":"Baudot","year":"2015","journal-title":"Entropy"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"58","DOI":"10.2307\/1969145","article-title":"On the cohomology groups of an associative algebra","volume":"46","author":"Hochschild","year":"1945","journal-title":"Ann. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"591","DOI":"10.2307\/1969740","article-title":"Cohomology of Lie algebras","volume":"57","author":"Hochschild","year":"1953","journal-title":"Ann. Math."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"McCleary, J. (2001). A User\u2019s Guide to Spectral Sequences, Cambridge University Press.","DOI":"10.1017\/CBO9780511626289"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1090\/S0002-9947-1949-0033001-0","article-title":"Cohomology of space with operators group","volume":"65","author":"Eilenberg","year":"1949","journal-title":"Trans. Am. Math. Soc"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"139","DOI":"10.24033\/asens.1263","article-title":"Homologie des complexes de formes diff\u00e9rentielles d\u2019ordre sup\u00e9rieur","volume":"7","author":"Koszul","year":"1974","journal-title":"Ann. Sci. Ecole Norm. Super."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Kass, R.E., and Vos, P.W. (1997). Geometrical Foundations of Asymptotic Inference, Wiley.","DOI":"10.1002\/9781118165980"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Moerdijk, I., and Mrcun, J. (2003). Introduction to Foliations and Lie Groupoids, Cambridge University Press. Cambridge Studies in Advanced Mathematics.","DOI":"10.1017\/CBO9780511615450"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Molino, P. (1988). Riemannian, Foliations, Birkhauser.","DOI":"10.1007\/978-1-4684-8670-4"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"119","DOI":"10.2307\/1970097","article-title":"Foliations with bundle-like metrics","volume":"69","author":"Reinhardt","year":"1959","journal-title":"Ann. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/s00029-004-0362-x","article-title":"Linearizability of d-webs, on two-dimensional manifolds","volume":"10","author":"Akivis","year":"2004","journal-title":"Sel. Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2643","DOI":"10.1016\/S0362-546X(01)00385-6","article-title":"On the linearization of 3-webs","volume":"47","author":"Grifone","year":"2001","journal-title":"Nonlinear Anal."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1016\/0040-9383(93)90004-F","article-title":"Sur la lin\u00e9arisation des tissus de C2","volume":"32","author":"Henaut","year":"1993","journal-title":"Topology"},{"key":"ref_44","unstructured":"Hess, H. (1980). Differential Geometry Methods in Maththematical Physics, Springer. Lectures Notes 836."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"61","DOI":"10.7146\/math.scand.a-12525","article-title":"Structures localement plates dans certaines vari\u00e9t\u00e9s symplectiques","volume":"76","year":"1995","journal-title":"Math. Scand."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/BF01458323","article-title":"De Rham-Hodge Theory for Riemannian foliations","volume":"277","author":"Kamber","year":"1987","journal-title":"Math. Ann."},{"key":"ref_47","unstructured":"Byande, P.M. (2012). De Structures Affines \u00e0 la G\u00e9om\u00e9trie de L\u2019information, \u00c9ditions Universitaires Europ\u00e9ennes. (In French)."},{"key":"ref_48","unstructured":"Chentsov, N.N. (1982). Statistical Decision Rules and Optimal Inference, American Mathematical Society."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Petrie, T., and Handal, J. (1990). Connections, Definite Forms and Four-Manifolds, Clarendon Press. Oxford Mathematcal Monograph, Oxford Science Publications.","DOI":"10.1093\/oso\/9780198535997.001.0001"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1090\/S0002-9904-1964-11019-3","article-title":"An algebraic model for transitive differential geometry","volume":"70","author":"Guillemin","year":"1964","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02787690","article-title":"The infinte groups and lie and Cartan","volume":"15","author":"Singer","year":"1965","journal-title":"J. Anal. Math. Jerus."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Shima, H. (2007). The Geometry of Hessian Structures, World Scientific.","DOI":"10.1142\/9789812707536"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Nielsen, F., and Barbaresco, F. (2015). Symplectic Structure of Information Geometry: Fisher Information and Euler-Poincare Equation of Souriau Lie Group Thermodynamics, Springer.","DOI":"10.1007\/978-3-319-25040-3_57"},{"key":"ref_54","unstructured":"Nielsen, F., and Barbaresco, F. (2013). Hessian Structures on Deformed Exponential Families, Springer."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"4521","DOI":"10.3390\/e16084521","article-title":"Koszul Information Geometry and Souriau Geometric Temperature\/Capacity of Lie Group Thermodynamics","volume":"16","author":"Barbaresco","year":"2014","journal-title":"Entropy"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/0021-8693(68)90004-5","article-title":"Commutative algebra cohomology and deformation of Lie algebras and associative algebras","volume":"9","author":"Nijenhuis","year":"1968","journal-title":"J. Algebra"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Chevallier, E., Barbaresco, F., and Angulo, J. (2016). Kernel Density Estimation on Siegel Space Applied to Radar Processing. Entropy, 18.","DOI":"10.3390\/e18110396"},{"key":"ref_58","unstructured":"Chevallier, E., Barbaresco, F., and Angulo, J. (2015). Geometric Science of Information, Springer."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1090\/S0002-9947-1947-0021313-4","article-title":"Homology of spaces with operators group I","volume":"62","author":"Eilenberg","year":"1947","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_60","unstructured":"Nguiffo Boyom, M., and Wolak, R. (2014, January 21\u201326). Transverse Hessian metrics information geometry. Proceedings of the 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Amboise, France."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Nguiffo Boyom, M., and Wolak, R.A. (2016). Transversely Hessian foliations and information geometry. Int. J. Math., 27.","DOI":"10.1142\/S0129167X16500920"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1017\/S0143385700000511","article-title":"On Gromov theory of rigid transformation groups, a dual approach","volume":"20","author":"Zeghib","year":"2000","journal-title":"Ergod. Theorey Dyn. Syst."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Kamber, F., and Tondeur, P.H. (1971). Invariant Operators and the Cohomology of Lie Algebra Sheaves, Memoirs of the American Matheamtical Society.","DOI":"10.1090\/memo\/0113"},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Kumpera, A., and Spencer, D. (1972). Lie Equation, Princeton University Press.","DOI":"10.1515\/9781400881734"},{"key":"ref_65","unstructured":"Guts, A.K. Private communication."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/12\/433\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:27:55Z","timestamp":1760210875000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/12\/433"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,2]]},"references-count":65,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2016,12]]}},"alternative-id":["e18120433"],"URL":"https:\/\/doi.org\/10.3390\/e18120433","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,12,2]]}}}