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.

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.

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.

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.

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 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. "

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.