Prof.dr.
N.P. Landsman, 01/09/2002-31/08/2007 (principal investigator)
Dr.
C. Carvalho, 01/09/2003-31/08/2005 (postdoc)
Drs.
P.Hochs, 01/04/2003-31/03/2007 (oio)
Drs.
N. Kowalzig , 01/09/2003-31/08/2007 (oio)
Dr.
M. MŸger, 01/07/2003-30/06/2006 (postdoc)
Drs.
W. van Suijekom, 01/09/2002-31/10/2002
Dr.
R. Popescu, 12/05/2003-10/06/2003
Drs.
E. van Erp, 17/05/2003-15/06/2003 and 11/11/2003-26/11/2003
So far, the project has largely been carried as
outlined in the proposal. A coherent team of scientists has been formed to investigate the
subprojects listed in the proposal, viz. Quantization of singular spaces (N. Kowalzig), Functoriality of
quantization (P.
Hochs and N. Kowalzig), Quantization and index theory (C. Carvalho), and Quantum symmetry (M. MŸger); see above
for details of their appointments. The principal investigator (PI) mainly
concerns himself with the first three themes, MŸger being a senior postdoc with
whom the fourth project is in good hands. All project members have built up an
impressive record of conference attendances (as detailed below) and scientific
activities in general, including attending and reporting on their work at the
PIÕs Seminar on Noncommutative
Geometry
at Amsterdam during the academic year 2002-2003 (joint with E.M. Opdam, see http://remote.science.uva.nl/~npl/BC.html)
as well as at his Seminar on Quantization, noncommutative geometry, and
symmetry
during 2003-2004 (see http://remote.science.uva.nl/~npl/group.html).
More generally, it should be mentioned that especially the oioÕs on the project
have learnt a great deal of new mathematics during the early phase of their
appointments.
Functoriality
of quantization. The
main achievement of the PI during the report period has been a precise version
of the conjectural functoriality of quantization: he proposes that geometric
quantization of symplectic manifolds is the arrow part of a functor, whose
object part is deformation quantization of Poisson manifolds. The Ôquantization
commutes with reductionÕ conjecture of Guillemin and Sternberg for compact Lie
groups acting on compact symplectic manifolds then becomes a special case of
the functoriality of quantization. In fact, this formulation yields almost
unlimited generalizations of the Guillemin--Sternberg conjecture, extending it,
for example, to arbitrary Lie groups or even Lie groupoids. Technically, this
involves symplectic reduction and Weinstein's dual pairs on the classical side,
and Kasparov's bivariant K-theory for C*-algebras (KK-theory) on the quantum
side. Thus the huge project Functoriality of quantization has now branched off
into two manageable subprojects, each probably worth a PhD thesis (hence the
role of the oioÕs involved):
i)
The Guillemin-Sternberg conjecture for noncompact Lie groups acting on
noncompact symplectic manifolds (PI and P. Hochs). This version of the
conjecture has now been formulated in a precise way, and has almost been proved
for the special case that the group in question has a discrete normal subgroup
with compact quotient group, whereas the group action is cocompact. This result
has been announced and discussed in a large number of talks by the PI (listed
below) in front of all relevant experts, such as Guillemin (at Utrecht) and
Weinstein (at Paris). Hochs and the PI are currently working on a publication
on this result, in which also various attractive examples will be added.
Consequently, the Guillemin-Sternberg conjecture, subject to a whole industry
already in the compact case, has now been freed of the limitation of
compactness, and has been proved in an important special case in the noncompact
setting.
Quantization of
singular spaces
N. Kowalzig started his appointment with the elaboration of his Diplomarbeit on the quantization of
regular symplectic quotients that are symplectic leaves of Poisson manifolds of
the type (T*P)/G (where a given manifold P is a G-space for some Lie group G).
This led to his first publication listed below. He is currently studying the
singular case, in which the G-action is not free. He attempts to combine the ideas in Xiang Tang, ÔDeformation
Quantization of Pseudo Symplectic(Poisson) GroupoidsÕ, arXiv:math.QA/0405378,
which uses the formalism of star-products or formal deformation quantization,
with those in the PIÕs paper ÔDeformation quantization and the Baum--Connes
conjectureÕ, Commun. Math. Phys. 237, 87-103 (2003), which uses
operator-algebraic techniques and noncommutative geometry: the quotient (T*P)/G
is desingularized in the spirit of Connes by the semidirect product groupoid
T*P x G. In the C*-algebraic framework used by the PI, quantization of both the
quotient and its desingularization is known. This appears to be a promising
strategy.
Quantization and
index theory.
In the past year C. Carvalho
carried out a research project in index theory in the framework of
almost differentiable groupoids. She spent some time getting a clear grasp on
the objects involved, namely on the general theory of pseudodifferential
operators on groupoids and on the significance of the analytic index in this
setting. For the time being, she has
been focusing mostly on groupoids associated to manifolds with corners.
In particular, she devoted some time understanding Melrose's b-calculus
approach to the Atiyah-Patodi-Singer index theorem on manifolds with boundary,
which somehow encapsulates the
general philosophy of the role of groupoids in index theory. In order to deal
with the more general case of manifolds with corners one should however replace
Melrose's heat kernel approach by different techniques, such as constructing
'shriek' maps suited to the b-tangent spaces (this seems to be work in process
by Monthubert and Nistor, with whom Carvalho (who is a former PhD student of
Nistor) might collaborate in the near future). She now plans to explore further
other settings where groupoids seem to play a significant role, such as
Atiyah's L2-index theorem, and to place it properly in the groupoid
framework, while still looking at the techniques used in the proof of the Connes-Skandalis
index theorem for foliations. Finally, she hopes to develop techniques to deal
with these index problems by combining the so-called 'shriek' methods with the
asymptotic morphisms approach of Higson to the classical index theorem; this
will involve using suitable quantisations of the algebras of symbols.
Quantum
symmetry. MŸgerÕs ongoing research revolves mostly about fusion categories, i.e., semisimple tensor
categories that are linear over a field, in relation to subfactor theory,
quantum field theory and representation theory, in particular categories and
the study of algebras in and module categories over such categories. In the
study of this area many subjects interact in a fascinating manner, and there is
a host of interesting structural and classification problems. In "On the
structure of modular categories" he proved a double commutant theorem for
modular categories and deduced that every modular category is a direct product
of prime ones. In "Galois extensions of braided tensor catgegories and
braided crossed G-categories", a sequel to his "Galois theory for braided tensor categories and the
modular closure", he proved that Galois extensions of braided tensor
categories give rise to braided crossed G-categories and studied the latter. These
results to answer several long-standing questions in quantum field theory
(CFT). His papers "On the center of a compact group" and
"Representations of algebraic quantum groups and reconstruction theorems
for tensor categories" are concerned with issues in the representation
theory of groups and quantum groups, respectively. In the first, MŸger proves
that the center of a compact group can be recovered from the representation
ring, and in the second, written jointly with L.Tuset and J.E.Roberts, he give
a streamlined presentation of the Tannaka reconstruction theory of discrete
quantum including some new results. MŸger is presently working on another joint
paper with L.Tuset "Regular representations of algebraic quantum groups
and embedding theorems", which will give a considerably simplified proof
of the abstract reconstruction theorem for compact groups, due originally to
P.Deligne and S.Doplicher/J.E.Roberts. "Extensions and modular invariants
of rational conformal field theories" will be
concerned with an approach to the
classification of 'modular invariants', which play a central role in the
construction of two-dimensional CFTs.
During 2002-2003 the PI organized two workshops
on topics central to the pionier project: a Workshop on the Quantization of
Singular Symplectic Quotients at Oberwolfach, 20-26/01/2003, and a Mini-Workshop on Noncommutative
Geometry, Amsterdam,
21-23/05/2003. With R. Sjamaar (Cornell) and J. Sniatycki (Calgary), he
prepared a proposal for a succesor to the first one, to be held at BIRS (Banff)
in 2005 or 2006 (application pending).
In
2002 he PI set up a Master Program in Mathematical Physics at the University of
Amsterdam, which should educate students in part for the kind of reserach in
this pionier project. During 2003-2004 the PIÕs main activity outside the
project was the MRI Masterclass on Noncommutative Geometry, organized jointly with
I. Moerdijk (Utrecht). This had a similar goal. With considerable investment of
time and energy, the PI supervised a PhD Thesis, as well as a number of M.Sc
Theses on topics related to the pionier project: ÔQuantization of Hamiltonian
loop group actionsÕ by H. Postuma (doctorate awarded 17/09/2003), and
ÔQuasicrystals, C*-algebras and K-theoryÕ by F. Ypma and ÔCoarse Geometry and
K-HomologyÕ by B.NorouzizadehÕ (the latter a Masterclass essay). Other project
members were active in teaching as well: MŸger taught courses on algebraic and
differential topology, Carvalho led the exercise class on C*-algebras and
K-theory during the above mentioned Masterclass, and Hochs led various exercise
classes in elementary mathematics during his appointment.
Spring
and Summer of 2004 were largely occupied with the preparation of the proposal The
Fellowship of Geometry and Quantum Theory for an NWO Wiskundecluster, whose
success or failure will become known shortly (ans probably is already known at
the time the GBE reads this report). If successful, about one third of the
research proposed in this cluster will be closely related to the present
pionier project, the remainder still being relevant. More generally, this
cluster would provide an enormous impetus to the areas of mathematical physics
and mathematics the PI is interested in.
As
may be inferred from his list of talks below, the PI spent considerble effort
on Outreach activities, addressing audiences from highschool students to
retiring ÔPR-medewerkersÕ. These activities culminated in his popular book
ÔRequiem for NewtonÕ to appear shortly (which, however, was entirely written
during evenings, weekends and holidays).
Finally,
the move of the PI and the pionier project from the UvA to the RUN took effect
on 01/09/2004 and took a longt ime to prepare. For reasons outlined in more
detail in his recent letter to the GBE, the PI expects the move to
significantly benefit the project, largely because of the reduction in travel
time as well as the presence of mathematicians such as Heckman and Maassen at
Nijmegen, whose expertise will undoubtedly be important to the project (this
expectation has been realized already, as Heckman is currently writing a
conference paper in collaboration with project member P. Hochs). Also, since
the RUN is part of MRI, the PIÕs ties with Crainic and Moerdijk at Utrecht will
be facilitated.
Nijmegen, 7 October, 2004
N.P. Landsman
N.P.
Landsman:
Refereed:
1.
Quantization as a functor. In: Quantization, Poisson Brackets, and Beyond, ed.
T. Voronov, Contemporary Mathematics 315, pp. 9-24 (AMS, Providence, 2002). arXiv:math-ph/0107023.
2.
Quantization and the tangent groupoid. In: Operator Algebras and Mathematical
Physics
eds.
J.-M. Combes, et al., pp. 251-265
(Theta Foundation, 2003). arXiv:math-ph/0208004
3.
Deformation quantization and the Baum-Connes conjecture. Communications in
Mathematical Physics 237, 87-103 (2003). arXiv: math-ph/0210015.
4.
Functorial quantization and the Guillemin--Sternberg conjecture. In: Proc. XXth
Workshop on Geometric Methods in Physics (Bialowieza, 2002), eds. S.T Ali et
al. arXiv:math-ph/0307059.
5.
Quantum mechanics and representation theory: the new synthesis. Acta
Applicandae Math. 81, 167-189 (2004).
Popular (in Dutch):
Wiskunde
aan de grenzen van de natuurkunde (oratie). Nieuw Archief voor Wiskunde 5/3,
24-32 (2002).
Wie
was Thomas Young? Nederlands Tijdschrift voor Natuurkunde 69, 40-44 (2003).
De
dramatiek van de natuurkunde: Toneelstukken van Frayn en Rijnders onder de
loep.
Kunst
en Wetenschap 12 (2), 5-8 (2003).
Briljante
gifmenger: een portret van Isaac Newton. Academische Boekengids 39 (juni),
13-14 (2003).
De
indexstelling van Atiyah en Singer. Nieuw Archief voor Wiskunde 5/5, 207-211
(2004).
Requiem
voor Newton (Uitgeverij Contact, Amsterdam, 2005, 400 p.).
C. Carvalho:
A
K-theory proof of the cobordism invariance of the index, arXiv.math.KT/0408260.
N. Kowalzig:
Phase
Space Reduction of Star Products on Cotangent Bundles. Ann. Inst. H. Poincare,
in press. arXiv:math.SG/0403239.
M. MŸger:
1.
On the structure of modular categories. Proc. Lond. Math. Soc. 87, 291-308
(2003).
2.
Galois extensions of braided tensor catgegories and braided crossed
G-categories. J. Alg. 277, 256-281 (2004).
3.
On the center of a compact group. Intern. Math. Res. Notes. 2004:51, 2751-2756
(2004)
4.
M. Mueger, L. Tuset, J.E. Roberts: Representations of algebraic quantum groups
and reconstruction theorems for tensor categories. Alg. Repres. Theor. (2004)
in press. arXiv:math.QA/0203206.
5.
Conformal orbifold theories and braided crossed G-categories. Commun. Math.
Phys. in press. arXiv:math.QA/0403322.
Project
guests:
R.
Popescu, Equivariant E-theory for groupoids acting on C*-algebras. J. Funct.
Anal. 209, 247-292 (2004).
W.
D. van Suijlekom, The noncommutative Lorentzian cylinder as an isospectral deformation. J. Math. Phys. 45, 537-556 (2004).
N.P.
Landsman:
2004:
Friday,
9 July, 16:00-17:00, Paris
(K-theory
and Noncommutative Geometry)
The ÔQuantization commutes with reductionÕ conjecture for noncompact
groups
Tuesday,
June 29, 20:00-22:00, Leusden
(Wiskunde
en Werkelijkheid)
Wiskunde
of Religie: Spinoza versus Newton
Wednesday,
June 23, 15:30-16:30. Utrecht
(MRI
Workshop on Lie Groups in Analysis, Geometry, and Mechanics)
The
Guillemin--Sternberg conjecture
for noncompact groups
Wednesday,
June 16, 15:00-16:00. Frankfurt
(Workshop
zu Indextheorie und Quantisierung)
Die
Vermutung von Guillemin--Sternberg
fŸr nichtkompakte Gruppen
Thursday,
May 27, 15:30-17:00, Utrecht
The
Atiyah-Singer index theorem (with J.J. Duistermaat)
Thursday,
13 May, 9:15-11:00, Amsterdam
(Highlights-College)
De
indexstelling van Atiyah en Singer voor eerstejaars verklaard
Wednesday,
12 May, 11:15-12:15. Amsterdam
The
Atiyah-Singer index theorem
Wednesday,
12 May, 15:15-16:00. Amsterdam
Wiskunde
en de Verlichting: Descartes - Spinoza - Newton
Tuesday,
30 March, 14:00-16:00, Amsterdam
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
Geometric
quantization III
Tuesday,
16 March, 14:00-15:00, Marseille-Luminy
(Geometric
Analysis)
Quantization, the analytic assembly map, and the
Guillemin--Sternberg conjecture for noncompact groups
Tuesday,
9 March, 14:00-16:00, Amsterdam
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
Geometric
quantization II
Tuesday,
17 February, 14:45-15:45, Marseille-Luminy
(Noncommutative
Geometry in Mathematics and Physics)
Functoriality
of quantization and the Guillemin-Sternberg conjecture
Tuesday,
3 February, 14:00-16:00, Amsterdam
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
Geometric
quantization I
2003:
Tuesday,
October 14, 14:00-16:00, Amsterdam
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
From
index theory to (geometric) quantization
Tuesday,
October 7, 14:00-16:00, Amsterdam
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
From
(deformation) quantization to index theory
Wednesday,
August 20, 10:00-17:00, Villigst
(Welle
trifft Teilchen)
Klassische
und Quantenmechanik
Thursday,
August 21, 20:00-22:00, Villigst
(Welle
trifft Teilchen)
Werner
Heisenberg und die deutsche Atombombe
Friday,
July 11, 15:00-16:00, Aachen
Quantum
mechanics and representation
theory: the new synthesis
Wednesday,
11 June, 16:00-16:30, Amsterdam (CWI)
Kansen
en valkuilen bij de popularisering van de wiskunde
Friday,
23 May, 15:30-16:30 Amsterdam
(Mini-Workshop
on Noncommutative Geometry)
Functoriality
of quantization, KK-theory, and the Guillemin-Sternberg conjecture
Wednesday,
21 May, 15:15-16:00, Amsterdam
Wiskunde
en de Verlichting: Descartes - Spinoza - Newton
Friday,
9 May, 14:00-14:45, Amsterdam
(Leve
de Wiskunde)
De
wiskunde achter de kwantumtheorie: Van atoomspectra tot niet-commutatieve
meetkunde
Thursday,
1 May, 11:00-11:45, Nijmegen
(Nederlands
Mathematisch Congres)
Quantum
mechanics and representation
theory: the new synthesis
Friday,
April 11, 11:00-13:00, Amsterdam
(Seminar
on Noncommutative Geometry)
The
general Baum-Connes conjecture
Friday,
4 April, 11:15-12:15, Utrecht
The
royal path to quantization
Tuesday,
11 March, 12:45-13:45, Delft
De
wiskunde achter de quantumtheorie: Van atoomspectra tot niet-commutatieve meetkunde
Wednesday,
5 March, 16:00-17:30, Amsterdam
Eerste
drukken van boeken van Newton, Huygens and Descartes in de UBA
Wednesday,
5 March, 20:00-22:00, Amsterdam
Uit
het raam staren (gesprek met Gerardjan Rijnders)
Thursday,
13 February, 15:30-17:30, Utrecht
Quantization
as a functor
Friday,
31 January, 11:00-11:45, Amsterdam
Isaac
Newton: de eerste moderne wiskundige
Thursday,
23 January, 10:00-12:00, Oberwolfach
(Workshop
on the Quantization of Singular Symplectic Quotients)
Quantization
as a functor
Wednesday,
15 January, 11:15-12:45, Hamburg
Quantization
as a functor
2002:
Friday,
6 December, 11:00-13:00, Amsterdam
(Seminar
on Noncommutative Geometry)
KK-theory
Wednesday,
13 November,14:00-15:00, Amsterdam
Isaac
Newton: de eerste mathematisch fysicus
Friday,
1 November, 14:00-16:00, Amsterdam
(Seminar
on Noncommutative Geometry)
Connes's
proof of the Atiyah-Singer index theorem II
Friday,
18 October, 11:00-13:00, Amsterdam
(Seminar
on Noncommutative Geometry)
Connes's
proof of the Atiyah-Singer index theorem I
Friday,
20 September, 11:00-13:00,
Amsterdam
(Seminar
on Noncommutative Geometry)
The Atiyah-Singer index theorem
Friday,
6 September, 11:00-13:00,
Amsterdam
(Seminar
on Noncommutative Geometry)
Overview:
from Atiyah-Singer to V. Lafforgue
C. Carvalho:
Index
theory on noncompact manifolds I,
Amsterdam, 21-10-03
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
Index
theory on noncompact manifolds II, UvA,
Amsterdam, 4-11-03
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
Index
theory for groupoids I, UvA,
Amsterdam, 18-06-04
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
Index
theory for groupoids II, RU, Nijmegen, 07-08-04
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
Index
theory for groupoids III, RU, Nijmegen, 14-08-04
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
P. Hochs:
Localisation at fixed point sets I, 11
november 2003
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
Localisation
at fixed point sets II, 2 december 2003
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
Symplectic
reduction as a map on K-homology I, 21 september 2004
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
Symplectic
reduction as a map on K-homology
II, 28 september 2004,
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
N. Kowalzig:
9
December 2003:, Reduction of star products on cotangent bundles I
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
18
December 2003, 14:15-16:00: Reduction of star products on cotangent bundles II
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
M. MŸger:
The
noncommutative-geometry approach to the Novikov conjecture I, 13 January 2004
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
The
noncommutative-geometry approach to the Novikov conjecture II, 20 January 2004
(Seminar
on Quantization, Noncommutative Geometry, and Symmetry)
The
idempotent conjecture and the conjectures of Novikov and Baum/Connes, Aachen,
November
2003 (6 hour lecture series)
Conformal
orbifold models and braided crossed G-categories, held at:
University
'Tor Vergata', Rome (May 2004)
Erwin
Schroedinger Institute, Vienna (Program on Tensor categories in mathematics and
physics 14.-26.06.2004)
University
of Hamburg: 14th Workshop on Foundations and Constructive Aspects of QFT 02.-03.7.2004)
N.P.
Landsman:
Workshop
on the Quantization of Singular Symplectic Quotients, Oberwolfach,
20-26/01/2003 (Organizer and speaker)
Mini-Workshop
on Noncommutative Geometry, Amsterdam, 21-23/05/2003 (Organizer and speaker)
Nederlands
Mathematisch Congres, Nijmegen, 1-2/05/2003 (Invited speaker)
Welle
trifft Teilchen, Villigst, 18-22/08/2003 (Invited speaker)
Noncommutative
Geometry in Mathematics and Physics, Marseille-Luminy, 16-20/02/2004 (Invited
speaker)
Geometric
Analysis, Marseille-Luminy, 15-19/03/2004 (Invited speaker)
Workshop
zu Indextheorie und Quantisierung, Frankfurt, 16/06/2004 (Invited speaker)
Wiskunde
en Werkelijkheid, Leusden, 28-31/06/2004 (Invited speaker)
MRI
Workshop on Lie Groups in Analysis, Geometry, and Mechanics, Utrecht,
21-25/06/2004 (Invited speaker)
K-theory
and Noncommutative Geometry, Paris, 5-9/07/2004 (Invited speaker)
C. Carvalho:
Non-commutative
Geometry in Mathematics and Physics: K-theory, CIRM Marseille, 9-02-04 to
13-02-04
Non-commutative
Geometry in Mathematics and Physics: Non-commutative Geometry in Physics, CIRM
Marseille, 16-02-04 to 20-02-04;
Geometric
Analysis: CIRM Marseille, Marseille, 15-03-04 to 19-03-04;
K-theory
and Non-commutative geometry, IHP Paris, 4-07-04 to 15-07-04.
P. Hochs:
The
Clay mathematics institute conference and spring school on noncummutative
geometry and applications, Vanderbilt university, Nashvill, Tennessee, VS, 2
Ð13/05/2003
School
and conference PQR 2003 on Poisson geometry, Quantization and group
Representations, Vrije universiteit Brussel, België, 13 Ð22/06/2003
Noncommutative
geometry in mathematics and physics, Centre International de Rencontres
Mathématiques, Marseille, Frankrijk, 9 - 20/02/2004
Spring
school Lie groups in analysis, geometry and physics, Universiteit Utrecht,
May-June 2004
N. Kowalzig:
Poisson
Geometry, Deformation Quantisation and Group Representations (Bruxelles June
2003, prior to appointment)
DMV-Tagung
(Rostock, September 2003)
K-Theory
and Cyclic Cohomology: Noncommutative Geometry in Mathematics and Physics
(Luminy/Marseille February 2004)
K-Theory
and Noncommutative Geometry School followed by a conference
the
same topics (Paris, May-June-July 2004, on unpaid leave
from the UvA).
M. MŸger:
25.-31.1.2004 Oberwolfach-Seminar: `The
Novikov Conjecture: Geometry
and Algebra'
08.-12.3.2004:
University of Muenster: Workshop `L2-Invariants,
Measure Theory and Geometric Group
Theory'
14.-26.06.2004:
Erwin Schroedinger Institute, Vienna: Program on
`Tensor categories in mathematics
and physics'
02.-03.7.2004:
University of Hamburg: 14th Workshop "Foundations and
Constructive Aspects of
QFT"
05.-16.7.2004:
Institute Henri Poincare', Paris: Conference
`K-Theory and Noncommutative
geometry'
19.-21.7.2004:
Saclay Paris: Symposium in honor of J. Bros `Rigorous Quantum Field Theory'