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This volume contains the proceedings of the Colloquium "Analysis,
Manifolds and Physics" organized in honour of Yvonne Choquet-Bruhat
by her friends, collaborators and former students, on June 3, 4 and
5, 1992 in Paris. Its title accurately reflects the domains to
which Yvonne Choquet-Bruhat has made essential contributions. Since
the rise of General Relativity, the geometry of Manifolds has
become a non-trivial part of space-time physics. At the same time,
Functional Analysis has been of enormous importance in Quantum
Mechanics, and Quantum Field Theory. Its role becomes decisive when
one considers the global behaviour of solutions of differential
systems on manifolds. In this sense, General Relativity is an
exceptional theory in which the solutions of a highly non-linear
system of partial differential equations define by themselves the
very manifold on which they are supposed to exist. This is why a
solution of Einstein's equations cannot be physically interpreted
before its global behaviour is known, taking into account the
entire hypothetical underlying manifold. In her youth, Yvonne
Choquet-Bruhat contributed in a spectacular way to this domain
stretching between physics and mathematics, when she gave the proof
of the existence of solutions to Einstein's equations on
differential manifolds of a quite general type. The methods she
created have been worked out by the French school of mathematics,
principally by Jean Leray. Her first proof of the local existence
and uniqueness of solutions of Einstein's equations inspired Jean
Leray's theory of general hyperbolic systems.
to our own also needs to be understood. Such unification may also
require that the supersymmetry group possess irreducible
representations with infinite reductiori on the Poincare subgroup,
to accommodate an infinite set of particles. Such possibilities
were 5 envisaged long ago and have recently reappeared in
Kaluza-Klein . 6 d' . th 7 S . l' th supergraVlty an m superstnng
eory. upersymmetry Imp Ies at forces that are mediated by bose
exchange must be complemented by forces that are due to the
exchange of fermions. The masslessness of neutrinos is
suggestive-we continue to favor the idea that neutrinos are
fundamental to weak interactions, that they will finally play a
more central role than the bit part assigned to them in
Weinberg-Salam theory. There seems to be little room for doubting
that supersymmetry is badly broken-so where should one be looking
for the first tangible manifestations of it? It is remarkable that
the successes that can be legitimately claimed for supersymmetry
are all in the domain of massless particles and fields.
Supergravity is not renormalizable, but it is an improvement (in
this respect) over ordinary quantum gravity. Finite super
Yang-Mills theories are not yet established, but there is now a
strong concensus that they soon will be. In both cases massless
fields are involved in an essential way.
This book presents the text of most of the lectures which were de-
livered at the Meeting Quantum Theories and Geometry which was held
at the Fondation Les Treilles from March 23 to March 27, 1987. The
general aim of this meeting was to bring together mathemati- cians
and physicists who have worked in this growing field of contact
between the two disciplines, namely this region where geometry and
physics interact creatively in both directions. It 1S the strong
belief of the organizers that these written con- tributions will be
a useful document for research people workin~ 1n geometry or
physics. Three lectures were devoted to the deformation approach to
quantum mechanics which involves a modification of both the
associative and the Lie structure of the algebra of functions on
classical phase space. A.Lichnerowicz shows how one can view
classical and quantum statistical mechanics in terms of a
deformation with a parameter inversely propor- tional to
temperature. S.Gutt reviews the physical background of star
products and indicates their applications in Lie groups representa-
tion theory and in harmonic analysis. D.Arnal gives a rigorous
theory Vll viii PREFACI of the star exponential in the case of the
Heisenberg group and shows how this can be extended to arbitrary
nilpotent groups.
This book contains the proceedings of a meeting that brought
together friends and colleagues of Guy Rideau at the Universite
Denis Diderot (Paris, France) in January 1995. It contains original
results as well as review papers covering important domains of
mathematical physics, such as modern statistical mechanics, field
theory, and quantum groups. The emphasis is on geometrical
approaches. Several papers are devoted to the study of symmetry
groups, including applications to nonlinear differential equations,
and deformation of structures, in particular
deformation-quantization and quantum groups. The richness of the
field of mathematical physics is demonstrated with topics ranging
from pure mathematics to up-to-date applications such as imaging
and neuronal models. Audience: Researchers in mathematical physics.
"
This book contains the proceedings of a meeting that brought
together friends and colleagues of Guy Rideau at the Universite
Denis Diderot (Paris, France) in January 1995. It contains original
results as well as review papers covering important domains of
mathematical physics, such as modern statistical mechanics, field
theory, and quantum groups. The emphasis is on geometrical
approaches. Several papers are devoted to the study of symmetry
groups, including applications to nonlinear differential equations,
and deformation of structures, in particular
deformation-quantization and quantum groups. The richness of the
field of mathematical physics is demonstrated with topics ranging
from pure mathematics to up-to-date applications such as imaging
and neuronal models. Audience: Researchers in mathematical physics.
"
This book presents the text of most of the lectures which were de-
livered at the Meeting Quantum Theories and Geometry which was held
at the Fondation Les Treilles from March 23 to March 27, 1987. The
general aim of this meeting was to bring together mathemati- cians
and physicists who have worked in this growing field of contact
between the two disciplines, namely this region where geometry and
physics interact creatively in both directions. It 1S the strong
belief of the organizers that these written con- tributions will be
a useful document for research people workin~ 1n geometry or
physics. Three lectures were devoted to the deformation approach to
quantum mechanics which involves a modification of both the
associative and the Lie structure of the algebra of functions on
classical phase space. A.Lichnerowicz shows how one can view
classical and quantum statistical mechanics in terms of a
deformation with a parameter inversely propor- tional to
temperature. S.Gutt reviews the physical background of star
products and indicates their applications in Lie groups representa-
tion theory and in harmonic analysis. D.Arnal gives a rigorous
theory Vll viii PREFACI of the star exponential in the case of the
Heisenberg group and shows how this can be extended to arbitrary
nilpotent groups.
Very few people have contributed as much to twentieth-century
physics as Julian Schwinger. It is therefore appropriate to offer a
retrospective of his work on the occasion of his sixtieth birthday
(February 12, 1978). We hope, in offering this selection of his
papers, to bring to light ideas and results that may have been
partly overlooked at the time of the original publication.
Schwinger has published prodigiously on a great variety of
subjects, as is evident from the comprehensive list of publications
arranged in chronological order which appears on p. xiii. Needless
to say, only a small subset could be included in the present modest
volume. In the selection, great weight was assigned to papers that
seem to be less widely known or appreciated than they deserve. Many
important papers are therefore omitted. (Examples: Paper [64] 'On
Gauge Invariance and Vacuum Polarization' and Paper [69] 'On
Angular Momentum', both of which have been reprinted elsewhere. )
The collection is a personal one, having been chosen by Schwinger
himself, and is therefore of particular interest. It would probably
not be interesting to offer an analysis, by the editors, of
Schwinger's contributions to physics. However, we are very pleased
to be able to include Schwinger's own informal and very personal
comments about each article that appears in this volume. These
comments indicate his reasons for choosing these particular
articles and, in many cases, provide a capsule synopsis of what he
considers most valuable.
Foreword. Introduction; J.C. Legrand. Relativistic dissipative
fluids; A.M. Anile, G. Ali, V. Romano. Mathematical problems
related to liquid crystals, superconductors and superfluids; H.
Brezis. Microcanonical action and the entropy of a rotating black
hole; J.D. Brown, J.W. York, Jr. Probleme de Cauchy sur un conoide
caracteristique. Applications a certains systemes non lineaires
d'origine physique; F. Cagnac, M. Dossa. Recent progress on the
Cauchy problem in general relativity; D. Christodoulou. On some
links between mathematical physics and physics in the context of
general relativity; T. Damour. Functional integration. A
multipurpose tool; C. DeWitt-Morette. Generalized frames of
references and intrinsic Cauchy problem in general relativity; G.
Ferrarese, C. Cattani. Reducing Einstein's equations to an
unconstrained hamiltonian system on the cotangent bundle of
Teichmuller space; A.E. Fischer, V. Moncrief. Darboux
transformations for a class of integrable systems in n variables;
C.H. Gu. Group theoretical treatment of fundamental solutions; N.H.
Ibragimov. On the regularity properties of the wave equation; S.
Klainerman, M. Machedon. Le probleme de Cauchy lineaire et
analytique pour un operateur holomorphe et un second membre
ramifie; J. Leray. On Boltzmann equation; P.L. Lions. Star products
and quantum groups; C. Moreno, L. Valero. On asymptotic of
solutions of a nonlinear elliptic equation in a cylindrical domain;
O. Oleinik. Fundamental physics in universal space-time; I. Segal.
Interaction of gravitational and electromagnetic waves in general
relativity; A.H. Taub. Anti-self dual conformal structures on
4-manifolds; C. Taubes. Chaotic behavior inrelativistic motion; E.
Calzetta. Some results on non constant mean curvature solutions of
the Einstein constraint equations; J. Isenberg, V. Moncrief. Levi
condition for general systems; W. Matsumoto. Conditions invariantes
pour un systeme, du type conditions de Levi; J. Vaillant. Black
holes in supergravity; P.C. Aichelburg. Low-dimensional behaviour
in the rotating driven cavity problem; E.A. Christensen, J.N.
Sorensen, M. Brons, P.L. Christiansen. Some geometrical aspects of
inhomogeneous elasticity; M. Epstein, G.A. Maugin. Integrating the
Kadomtsev-Petviashvili equation in the 1+3 dimensions via the
generalised Monge-Ampere equation: an example of conditioned
Painleve test; T. Brugarino, A. Greco. Spinning mass endowed with
electric charge and magnetic dipole moment; V.S. Manko, N.R.
Sibgatullin. Equations de Vlasov en theorie discrete; G. Pichon.
Convexity and symmetrization in classical and relativistic balance
laws systems; T. Ruggeri.
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