Wolfgang Bertram

Institut Elie Cartan de Lorraine
 Université de Lorraine
Faculté des Sciences et Techologies


accueil / welcome
responsabilités administratives et collectives
enseignement / teaching
doctorants / phd-students
activité de recherche / research activities
liste de publications scientifiques / list of scientific papers
sujets de recherche / research topics
problems, conjectures, speculations
divers / various stuff

Institut Elie Cartan de Lorraine
Site de Nancy
B.P. 70239
F - 54506 Vandoeuvre Cedex

bureau 410
(4e étage bâtiment IECL)  

Technical remark: the location of this page has changed recently, and some internal links won't work yet. I hope to get this fixed as soon as I have some time - sorry for this inconvenience!

accueil / welcome

Welcome on my personal homepage ! I apologize to French visitors that this page is not in French (and to German readers that it is not written in my mother language either). You can reach me under the paper mail address or phone number above, and - best - by mail under the address firstname.name (Klammeraffe) univ-lorraine.fr.

From 1999 to 2011, I have been  full professor of mathematics at Université Henri Poincaré  - Nancy I, and from 2012 on, at Université de Lorraine (which arose from a governement controlled fusion of the former universities of Nancy and Metz). Before taking the position in Nancy, I was Assistant (C1) at the Insitut fuer Mathematik at Technische Universitaet Clausthal (1996 - 1999), where I obtained my Habilitation in 1999. I spent a year (1995/96) at the Mittag-Leffler Institut in Djursholm near Stockholm, after having defended my phd-thesis "Dualité des espaces symétriques et analyse harmonique" at University Paris 6, under supervision of professor Jacques Faraut. I did my undergraduate and graduate studies (1985-1990) at Georg-August Universitaet Göttingen, with "Diplom Mathematik" under supervision of Horst Holdgrün and Gestur Olafsson. 

The French prime minister announced on november 19, 2018, a dramatic raise of student fees for foreign students. People from our university fight to prevent our government from realizing these plans - please join us !

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responsabilités administratives et collectives

2005 - janvier 2013 : responsable du Département de Formation Doctorale (DFD) mathématiques de l'Ecole Doctorale IAEM Lorraine

2005 - 2010 : responsable du Master 2 recherche mathématiques à Nancy. J'ai démissionné car j'ai jugé impossible de continuer cette responsabilité sous l'impératif de la réforme dite "Mastérisation".

du 1er février 2013 au 31 janvier 2017 : directeur du département de mathématiques, université de Lorraine, site de Nancy

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enseignement / teaching

Some remarks on teaching mathematics: In these short texts I present some classical topics in a new way, hopefully useful for teaching maths. The topics are related to my research - see the note given below for more ample comments. And see here what Vladimir Igorevitch Arnold said about teaching maths in France, some 20 years ago.
  1. Some remarks on teaching maths: differential calculus
  2. Some remarks on teaching maths: affine spaces and affine algebra
  3. Some remarks on teaching maths: the universal space
  4. Some remarks on teaching maths: torsors and principal equivalence relations
Lecture Notes (in english): "Concepts géométriques" ("Concepts in geometry", lectures given to phd-students, Nancy, spring 2016)

Enseignement 2020/21 :

Initiation à la recherche. From 2005 to 2010, our first year master program included a "student seminar", following the model of German universities. There were two different groups, one in pure and another in applied mathematics. In pure mathematics, we have studied the following texts:
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doctorants / phd-students

Manon DIDRY: thesis "Stuctures algébriques sur les espaces symétriques", defense june 16, 2006. Published in paper [24], below, and in: Construction of groups associated to Lie- and Leibniz-algebras (Journal of  Lie Theory 17 (2007), 399-426).

Julien CHENAL: thesis "Géométries liées aux algèbres de Lie graduées", defense june 21, 2010. Published in:
Arnaud SOUVAY: thesis "Une approche intrinsèque des foncteurs de Weil", defense november 23, 2012. Published in paper [30], below.

Jérémy HAUT: from fall 2018 on. Our joint paper (no [45] below) is accepted for publication in: Cahiers de topologie et géométrie différentielle catégoriques
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activités de recherche / research activities

I am member of the research team "Analyse et théorie des nombres" of the Institut Elie Cartan de Lorraine. Before 2013, I belonged to the team "Groupes de Lie et Analyse Harmonique" of the Institut Elie Cartan de Nancy. The team consists of researchers both from Metz and from Nancy, working on topics related to Lie groups and non-commutative harmonic analysis, operator algebras and geometry (non-commutative-, Poisson- or other), and on (mostly analytic) number theory. The seminars run on thursday afternoon, sometimes in Nancy and sometimes in Metz. About twice a year people from the east of France working on Lie groups and related topics meet for two days at the "Journées SL_2 R" (SLLR: Strasbourg-Lorraine-Luxembourg-Reims). I organized the meetings in Nancy in november 2004 and june 2006. The extended meeting to the honor of Jacques Faraut's 80th birthday, scheduled for march 12 and 13, 2011, had unfortunately to be cancelled due to the Covid-19 crises starting exactly at this moment.

From 2002 to 2011 I was (co-)organizer of the seminar "Groupes de Lie et Analyse Harmonique" in Nancy. Here are the links to the seminar programs of these years:
2010/11    2009/10    2008/09    2007/08   2006/07    2005/06    2004/05   2003/04   2002/03

I was involved in the organization of two international conferences:
Parallel to our seminar, I have organised some "Groupes de travail" in Nancy:

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  liste de publications / list of papers

See here for a pdf-file containing my publication list with hyperlinks, and see below for a description of my resarch topics.

  1. Realisierung einer holomorphen diskreten Reihe auf dem einschaligen Hyberboloid. Diplomarbeit, Goettingen 1990
  2. Dualité des espaces symétriques et analyse harmonique. Thèse de doctorat (phd-thesis), supervised by professor Jacques Faraut, université Paris VI, 1994
  3. The Geometry of Jordan and Lie Structures. Habilitationsschrift, Technische Universitaet Clausthal, 1999

Books :
  1. The Geometry of Jordan and Lie Structures (Springer Lecture Notes in Mathematics vol. 1754) - see above Habilitationsschrift for essentially the same text!
  2. Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, (186 + v pages),  Memoirs of the AMS 192, no.900 (2008)  (cf. article 19 ci-dessous)
  3. Calcul différentiel topologique élémentaire, Calvage et Mounet, Paris, décembre 2011  (lien fiche)

Papers :

  1. Généralisation d'une Formule de Ramanujan dans le cadre de la dualité des espaces Riemanniens symétriques.  Comptes Rendus Acad. Sci. Paris, t. 316 (1993), Série I,  1161-1166.
  2. Les Formules de Mehler et de Heine généralisées pour les espaces riemanniens symétriques de rang un. Comptes Rendus Acad. Sci. Paris   t.  318 (1994),  111-116.
  3. Un théorème de Liouville pour les algèbres de Jordan.  Bull. Soc. Math. Francaise 124 (1996), 299-327. pdf 
  4. On some Causal and Conformal Groups. J. Lie Theory 6 (1996), 215-244. pdf 
  5. Ramanujan's master theorem and duality of symmetric spaces. J. of Funct. An. 148 (1997), 117-151.pdf
  6. Algebraic Structures of Makarevi\v c Spaces. I. Transformation Groups, Vol. 3, No.1, (1998), 3-32. pdf
  7. Conformal group and fundamental theorem for a class of symmetric spaces. Math. Z.  233 (2000), 39 -73. pdf
  8. Reproducing kernels on vector bundles.  In: Lie Theory and Its Applications in Physics III, p. 43 - 58. World Scientific, Singapore 1998.  Avec J. Hilgert.  pdf
  9. Hardy Spaces and Analytic Continuation of Bergman Spaces. Bull. Soc. Math. Francaise 126 (1998), 435-482. Avec J. Hilgert.  pdf
  10. Geometric Bergman and Hardy spaces. Michigan Math. J. 47 (2000), 235 -263 Avec  J. Hilgert.   pdf
  11. Complexifications of Symmetric Spaces and Jordan Theory. Transactions of the A.M.S. 353 (2001), 2531 - 2556    dvi
  12. Characterization of the Kantor-Koecher-Tits algebra by a generalized Ahlfors operator. J. of Lie Theory 11 (2001), 415-426. Avec J. Hilgert.   pdf
  13. Generalized projective geometries: From linear algebra via affine algebra to projective algebra. Linear Algebra and its Applications 378 (2004), 109 - 134.    pdf
  14. Generalized projective geometries: General theory and equivalence with Jordan structures. Advances in Geometry 3 (2002), 329-369.   pdf
  15. The geometry of null systems, Jordan algebras and von Staudt's Theorem. Ann. Inst. Fourier 53 (2003) fasc. 1, 193-225. pdf
  16. Complex and quaternionic structures on symmetric spaces - correspondence with Freudenthal-Kantor triple systems. In : Theory of Lie Groups and Manifolds, Sophia Kokyuroku in Maths. 45 (2002), 57-76.  pdf
  17. Differential Calculus over general base fields and rings. (avec H. Gloeckner et K.-H. Neeb),  Expo. Math. 22 (2004), 213-282     pdf  arXiv : math.GM/0303300
  18. Projective completions of Jordan pairs. Part I: The generalized projective geometry of a Lie algebra (with K.-H. Neeb),  J. of Algebra 227 , 2 (2004), 474-519   pdf  arXiv : math.RA/0306272
  19. Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings. (186 + v pages),  Memoirs of the AMS 192, no.900 (2008)    pdf  arXiv math.DG/0502168
  20. Projective completions of Jordan pairs. Part II: Manifold structures and symmetric spaces (with K.-H. Neeb), Geometriae Dedicata 112 , 1, (2005), 73-113.    pdf    math.GR/0401236 
  21. Inner Ideals and Intrinsic Subspaces. (Avec H. Loewe.)  Adv. in Geometry 8 (2008), 53-85 ps math.RA/0606448
  22. Homotopes and conformal deformations of symmetric spaces.     J. of Lie Theory 18 (2008), no.2, 301-333    pdf    math.RA/0606449
  23. Is there a Jordan geometry underlying quantum physics?  Int. J. of Theoretical Physics 47 (no. 2) (oct. 2008), 2754-2782    pdf     math-ph/0801.3069
  24. Symmetric bundles and representations of Lie triple systems (avec M. Didry) Journal of Generalized Lie Theory and Applications 3 (no.4) (2009), 261-284   pdf   math.DG/0710.1543
  25. Associative Geometries. I: Torsors, Linear Relations and Grassmannians (avec M. Kinyon)  Journal of Lie Theory 20 (2) (2010), 215-252     pdf   : math.RA/0903.5441
  26. Associative Geometries. II: Involutions, the Classical Torsors, and their Homotopes (avec M. Kinyon)  Journal of Lie Theory 20 (2) (2010), 253-282   pdf    math.RA/0909.4438
  27. Simplicial differential calculus, divided differences, and construction of Weil functors.  pdf    Forum Mathematicum 25 (1) (2013), 19-47     arxiv :  math.DG/1009.2354
  28. Homotopes of symmetric spaces. I : Construction by algebras with two involutions.      pdf  (with P. Bieliavsky). to appear    math.DG/1011.2923
  29. Homotopes of symmetric spaces. II : Structure Variety and Classification      pdf   (Avec P. Bieliavsky). to appear ;      math.DG/1011.3161
  30. A general construction of Weil functors. (with A. Souvay) Cahiers de topologie et géométrie différentielle catégoriques LV (4) (2014), 267 -- 313,   pdf     math.DG/1009.2354
  31. The projective geometry of a group.   pdf      arxiv :   math.GR/1201.6201
  32. Torsors and ternary Moufang loops arising in projective geometry (avec M. Kinyon). p  343 - 360 in: Algebra, Geometry and Mathematical Physics, Springer-Verlag 2014 (Proceedings of the AGMP, Mulhouse, France, October 2011)    pdf    http://arxiv.org/abs/1206.2222
  33. Commutative and non-commutative parallelogram geometry: an experimental approach    pdf      http://arxiv.org/abs/1305.6851
  34. Jordan Geometries - an Approach via Inversions. Journal of Lie Theory 24 (2014) 1067-1113    pdf   http://arxiv.org/abs/1308.5888
  35. Weil Spaces and Weil-Lie groups.     http://arxiv.org/abs/1402.2619
  36. Universal Associative Geometry.     http://arxiv.org/abs/1406.1692
  37. Conceptual Differential Calculus. I : First order local linear algebra.    http://arxiv.org/abs/1503.04623
  38. Conceptual Differential Calculus. II : Cubic higher order calculus.      http://arxiv.org/abs/1510.03234
  39. A precise and general notion of manifold.     http://arxiv.org/abs/1605.07745
  40. Lie Calculus.  Proceedings of 50. Seminar Sophus Lie, Banach Center Publications 113 (2017), 59-85       https://arxiv.org/abs/1702.08282
  41. Cyclic orders defined by ordered Jordan algebras   Journal of Lie Theory 28 (3) (2018), 643-661  https://arxiv.org/abs/1706.09155 
  42. An essay on the completion of quantum theory. I: General setting.   https://arxiv.org/abs/1711.08643
  43. An essay on the completion of quantum theory. II: Unitary time evolution.   https://arxiv.org/abs/1807.04650  (Both parts dedicated to the memory of Tobias Brandes.) 
  44. Distributive lattices, and associative geometries: the arithmetic casehttps://arxiv.org/abs/2006.04450
  45. A functorial approach to differential calculus, https://arxiv.org/abs/2006.04452 (avec Jérémy Haut)
  46. Graded sets, graded groups, and Clifford algebras, https://arxiv.org/abs/2109.00878v1

Other papers (proceedings, overviews, lecture notes):

  1. Jordan algebras and conformal geometry.  In Positivity  in Lie Theory: Open Problems. (p. 1 - 20) de Gruyter, Berlin 1998.
  2. From Vector spaces to Symmetric Spaces. In : Lie Theory and its Applications in Physics, III (p. 99 - 109), World Scientific, Singapore 2000.
  3. Symmetric spaces with Jordan structures. In: Banach Center Publications 55 (2002),   211-226. ps
  4. Generalized projective geometries.  In: An. Univ. din Timisoara Vol. XXXIX, 2001 (Proceedings Fifth International Workshop on Differential Geometry and  Its Applications). pdf 
  5. Differential Geometry over General Base Fields and Rings. In: Modern Trends in Geometry and Topology, p. 95 - 102.  (Proceedings Seventh International Workshop on Differential Geometry and Its Applications, Cluj University Press 2006)    pdf
  6. Difference Problems and Differential Problems. In: Contemporary Geometry and Topology and  Related Topics, p. 73 - 86 (Proceedings Eighth International Workshop on Differential Geometry and Its Applications, Cluj University Press 2008)  pdf   arXiv:  math.GM/0712.0321
  7. Jordan structures and non-associative geometry. p. 221 - 241 in : Trends and Developments in Infinite Dimensional Lie Theory (ed. K.-H. Neeb and A. Pianzola),  Progress in Math. vol. 288, Birkhaeuser, New York 2011    pdf   arXiv:  math.RA/0706.1406
  8. On the Hermitian projective line as a home for the geometry of Quantum Theory. In:AIP Conference Proceedings 1079, p. 14 - 25 (Proceedings XXVII Workshop on Geometrical Methods  in Physics, Bialowieza 2008), American Institute of Physics, New York 2008   pdf    arXivmath-ph/0809.0561
  9. Jordan and Lie Geometries. Archivum mathematicum 49 (2013), 275 - 293. Notes of lectures given at the 33rd Winter School Geometry and Physics in Srni, january 2013.  pdf

Book review:
  1. "A Taste of Jordan Algebras" by K. McCrimmon (in: SIAM Review Vol. 47, No. 1 (2005), pp. 172-174 ) 

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sujets de recherche / research topics

Research Topics, and some Remarks on Teaching Mathematics (pdf-file containing hyperlinks)

The aim of these notes is to explain some of my research topics to non-specialists. I shall try to keep the text as non-technical as possible - it is not meant to be a scientific communication, but rather a personal and informal conversation with a reader supposed to be interested in mathematics and in the way mathematicians think. Thus I will allow myself to give some comments on related topics, such as teaching mathematics, and to speak about my personal experience and motivation. In chronological order, my research developed along the following three strands, which I shall explain in inverse chronological order:
  • non-commutative harmonic analysis (section 3),
  • the geometry of Jordan-, Lie- and associative structures (section 2),
  • general differential calculus (section 1).
Contents (see pdf for full text); I refer to the list of my papers as given above: research papers are labelled [n], books [Bn], theses [Tn], and others - proceedings, overviews, lecture notes - labelled [On]. 
1. Differential calculus. References: [B2], [B3], main papers: [17, 19, 37 40] , other papers: [27, 30, 35, O5, O6]
1.1. Introduction: some personal remarks and acknowledgments.
1.2. Topological differential calculus. [B2], [B3], [17]
1.3. Digression on linearity of differentials. [B3]
1.4. Conceptual differential calculus. [17, 37, 38, 40 B3, O6]
1.5. Weil functors. [B2, 27, 30, 35], thesis of Arnaud Souvay
2. Geometry and algebra. [B1], [3, 4, 7, 11, 13, 14, 15, 16, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 31, 32, 33, 36], [O1, O2, O3, O4, O7, O8, O9]
2.1. Jordan algebras: from physics to maths, from Goettingen via Paris to Clausthal. [B1], [3, 4, 6, 7, 11]
2.2. The general coquecigrue problem (hommage à Jean-Louis Loday)
2.3. Associative algebras and associative geometries. [25, 26, 32], [9]
2.4. Digression on non-commutative geometry. [O9]
2.5. Geometry at last! [32, 33]
2.6. Exotic planes. [32]
2.7. Classical spaces, Grassmannians and "relational mathematics". [31, 36]
2.8. Jordan geometries. [B1], [13, 14, 15, 21, 34, 41], [O1, O3, O4, O7, O9]
2.9. A "Jordan dictionary". [15, 21, O4, O7]
2.10. Models for Jordan geometries. [16, 18, 20], [O7, O9], thesis of Julien Chenal
2.11. Symmetric spaces. [B1, B2], [11, 22, 28, 29], thesis of Manon Didry
2.12. Liouville type theorems. [T2], [B1], [3, 4, 7, 15], [O1, O3]
2.13. Back to physics? [20, 23, 42, 43], [O8]
3. Harmonic analysis. [T1, T2], [1, 2, 5, 8, 9, 10, 12]
3.1. Goettingen
3.2. Paris
3.3. Clausthal
4. On teaching and research
4.1. Linear and affine algebra and geometry. [13, 33], [O2, O9]; notes on affine spaces, universal space and torsors, above.
4.2. Differential and integral calculus. [B2, B3]; note on differential calculus above.
5. On mathematics in France.

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problems, conjectures, speculations

Problems, Conjectures, Speculations  (pdf-file containing hyperlinks)

This is a kind of prolongation of the description of my research topics given above. I try to formulate some problems, conjectures and speculations that I find interesting (but most of them completely outside the scope of today's mainstream mathematics). Comments are welcome.

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divers / various stuff

some mathematical links:
some links concerning the state of research and universities in France and in the world:

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