GEOMETRY, ANALYSIS AND MATHEMATICAL PHYSICS EUROCONFERENCE ‘Analysis and Geometry’ 4-9 June 1999, Obernai FRANCE INTRODUCTION The relation of physics to mathematics is one of the most tantalising questions in pure science, and is the source of many debates. Until the beginning of the twentieth century, the two sciences seemed, at first sight, difficult to distinguish. Classical mechanics could hardly be described as mathematics or physics, but represented a fusion of the experimental and formalistic points of view. A new era was opened by Einstein, Hilbert and Poincaré. Poincaré was an extraordinary creative mathematician, whose influence on physics and mathematics was dramatic, from celestial mechanics to his contribution to relativity, and apparently at ease in both languages and both worlds. For Hilbert, for whom physics and mathematics were distinct bodies of knowledge, physical ideas had to be put on a sound conceptual and mathematical basis, and derived from first principles. Einstein not only revolutionised physics, but gave a dramatic impulse to mathematics by incorporating the geometric concepts elaborated by Riemann, Levi-Civita, Cartan as a central piece of general relativity. Einstein’s relativity synthesizes all the ambiguities in the relations of mathematics to physics. For the mathematicians of Einstein’s time, seeing geometrical ideas, which had been elaborated for no other reason than their internal beauty, put at work to explain the bending of light rays, the perihelion of Mercury, was certainly a dramatic development, largely unexpected. Since that time, it seems that geometry has been the hot point of the interaction between the two sciences, at a time where one could hardly claim to be a mathematician and a physicist. Quantum mechanics, the theory of small scale interactions in physics, has had an enormous influence on mathematics: quantization has become a standard name in mathematics. The theory of fibre bundles and connections, developed by mathematicians, appeared in a different language in physics through Yang-Mills theory, to the bewilderment of those involved. One should not draw the conclusions that the two sciences are becoming indistinguishable. Their relationship is one of attraction and conflict, interaction and repulsion.

The scope of the conference organised at Obernai was “Analysis and Geometry” as part of the series of conferences on “Analysis and mathematical Physics”. I will illustrate briefly one of the main topics of this conference, symplectic geometry. One can say that symplectic geometry developed first as the theory of canonical transformations in classical mechanics, a very clever way of solving complicate nonlinear equations in classical mechanics by nonlinear transformations. On a space, one can put rigid structures like Riemannian metrics, or flabby structures like symplectic structures. Contrary to metric spaces, symplectic spaces are all the same at small scale. Looking for invariants which may distinguish them at large scale has been a crucial questions in geometry. Here again, the relations of mathematics and physics catch the mind. Gromov discovered the roles of pseudoholomorphic curves, which play the role of geodesics in Riemannian geometry, which paved the way the construction of new global symplectic invariants. On the other hand, physicists, in their quest for grand unification, discovered that the collection of such invariants had surprising and largely unexpected properties. The purpose of the conference was to present recent developments illustrating interactions between physics and mathematics, in particular regarding symplectic geometry, Gromov-Witten invariants, moduli spaces, and Seiberg-Witten invariants. This is an extremely active area of mathematics, where surprising ideas and predictions coming from physics have led to a number of mathematical conjectures, whose verification represent a still immense task. In my opinion, the conference was a tremendous success. The sheer breadth and variety of points of view coming from the various talks was overwhelming, and represented much more one can expect from a specialised conference. The fact that many lecturers were young PhD from all countries, including from EEC or EEC affiliated countries, was also especially impressive. As an interested participant, I must say it was one of the most exciting conferences I have ever been to.

REPORT I will now try to report briefly on various aspects, which were illustrated during the conference, by mentioning some of the talks, which especially struck me. Symplectic geometry was present in many talks. Seidel’s lecture illustrated surprising connections between constructions in algebraic geometry involving the quotient of C2 by a cyclic group and other constructions involving Lagrangian submanifolds and Floer homology. Auroux’s lecture was about the construction of symplectic fibrations on a 4-manifold. These fibrations are associated to sections of high powers of the line bundle whose first Chern class is the corresponding symplectic form. This was illustrated in the exposition of joint work with Katzarkov, where in particular the branching locus of the fibration appears as a symplectic invariant. Yaël Karshon described the structure of manifolds equipped with a

torus action of complexity one, i.e. whose generic symplectic quotient is a Riemann surface. Polterovitch illustrated the interplay between symplectic topology and ergodic theory. Quantum cohomology appeared in various aspects. G. Liu gave his proof (joint work with Tian) of the Weinstein conjecture, using Gromov-Witten invariants. Givental showed how the computation of Riemann-Roch numbers on moduli spaces of holomorphic maps with values in a Grassmann manifold leads to a quantum version of the Toda lattice. Getzler gave a survey of the Virasoro conjecture, which has produced correct predictions for certain intersection numbers of moduli spaces of curves. Yau described his recent results on counting curves on algebraic manifolds, in particular for balloon manifolds. Chekhanov explained his construction of invariants of Legendrian knots using differential graded algebras. Giroux gave a survey of his description of contact structures. Meinrenken described his recent work on a highly non-trivial extension of classical equivariant cohomology to group values cohomology. In his construction, the symmetric algebra of the Lie algebra is replaced by the enveloping algebra, and the exterior algebra by the Clifford algebra. Seiberg-Witten invariants were extensively discussed by Pidstragatch, Feehan and Mrowka. Recent progress on the Witten conjecture relating Donaldson and Seiberg-Witten invariants were extensively discussed. Hutchings explained his recent work on the complexes associated to Morse functions on 3 dimensional manifolds and their relation to 3-dimensional Seiberg-Witten equation. The mathematics of M theory was explained by Dijkgraaf, and certain mathematical aspects of string theory were rigorously discussed by Nahm. On the more analytic side, Kuksin described a general framework for the treatment of elliptic PDE for maps values in manifolds. Karen Uhlenbeck discussed the adiabatic limit of the PDE characterising stable bundles when taking the adiabatic limit of the considered Kähler metric. It should be clear from the previous description that all subjects were seen from a varied and broad perspective. By bringing together some of the best European and US experts, by the important participation of young mathematicians from all over the worlds, the conference has, I think, been very useful to all involved. Jean-Michel Bismut, at Orsay, July 3rd 1999.

‘Analysis and Spectral Theory’ 22-27 September 2000, San Feliu de Guixols, SPAIN INTRODUCTION In this conference, we brought together a number of researchers in various closely interacting areas of spectral theory, with connections to analysis, geometry, dynamical systems and quantum physics. Most of the participants and speakers were mathematicians (or working with mathematically rigorous methods) while some where physicists. A number of very interesting recent results were presented, including: sharp spectral asymptotics for equations with non-smooth coefficients, upper bounds on the life span of metastable states, behaviour of non-linear evolution equations, spectral structure of Andersson Hamiltonians. The invited physicists presented interesting and challenging results for mathematicians to understand in depth about localisation, chaos and eigenvalue statistics. For administrative reasons the conference was divided into sessions occupying one or two half-days. With 3 or 4 exceptions the talks really fitted into this division, but because of late replacements of some speakers and last minute improvisations, due to early departures in the last days, complete agreement was impossible to attain. Below we discuss the contributions according to the categories where they actually belong.

REPORT 1) Control theory, spectral theory on manifolds and miscellaneous. (E. Zuazua, S. Zelditch, V. Ivrii, Maria Esteban, B. Helffer, J. Toth, A.Laptev.) The talks were devoted to quite a variety of subjects, localization and observation of waves in heterogeneous media, inverse spectral problems on domains with boundary and on manifolds, a new variation principle for finding the eigenvalues for the Dirac operator with numerical applications to quantum chemistry, the role of the boundary for the ground state of magnetic Schroedinger operators in domains with boundary, and general sharp spectral asymptotics for equations with non-smooth coefficients. One talk was about trace formulas and their relations to spectral results. 2) Non-linear scattering. (G. Velo, J. Ginibre, G. Perelman, V. Buslaev, A. Komech.) Two talks described the scattering theory of Hartree-type equations. Other recent results were about the behaviour near blow up for critical non-linear Schroedinger equations, soliton type solutions for equations with radiation, and a general approach to solvable models (including 1d models of statistical physics). 3) Resonances. (N. Burq, G. Vodev.) Here we regret the fairly late cancellation from one of the invited speakers. Both talks discussed more or less directly the fundamental question of upper bounds for lifetimes of metastable states, either in the case of the long-range perturbation of the Laplacian or in the case of certain manifolds of product type near infinity.

4) Many body theory. (J. Derezinski, J.P. Solovej, C. Ge'rard, C.A. Pillet, A. Vasy.) Pauli-Fierz equations were discussed in two talks, in connection with scattering theory and statistical physics. Another talk describes a general spectral approach to non-equilibrium statistical physics. One talk described a rigorous use of Bogoliubov's method for the asymptotics of the ground state energy for a system of a large number of bosons. An interesting geometrical approach to manybody quantum scattering was also presented. 5) Quantum chaos. (B. Eckhardt, U. Smilansky, A. Knauf, B. Altshuler, V. Jaksic.) Here three physicists presented results of wide interest. One talk was a very wide and clear survey of the current physicists' view on Andersson localization, quantum chaos and integrability and eigenvalue statistics. Another talk was about the general semiclassical approach to trace-formulas involving timeevolution for times currently beyond mathematically rigorous methods, and the third talk was about an intriguing duality between the interior and the the exterior for magnetic Schroedinger equations in plane domain. Rigourous results were also present in two of the talks, about the classical dynamics for many center problems, and for the spectral structure of Anderson type Hamiltonians.

SCIENTIFIC HIGHLIGHTS This only represent the chairman's view and other participants probably would like to emphasize other contributions. I think that the most impressive and important result presented were the sharp spectral asymptotic results of V. Ivrii for operators with non-smooth coefficients. They will probably be a main reference for a long time and they mark an important (perhaps final?) step in a long evolution started by H. Weyl in the beginning of the 20th century. Other highlights are in my opinion the results of N. Burq on the very general exponential lower bounds on the imaginary parts of resonances for general long range operators, and the result of J.P. Solovej and E. Lieb on the ground state energy for many-boson systems, where the taming of Bogoliubov's method seems important. Jaksic and Last's results on Andersson Hamiltons also look very interesting even though I am unable to predict exactly how far-reaching they will prove to be. The three physicist's talk where also of precious value for the mathematicians. Here the one of B. Eckhardt was perhaps the one which I felt was the most directly at the heart of the matter for people working with general semi-classical methods, while the survey of B. Altshuler gave a clear and broad overview, and the one of U. Smilansky described challenging numerical results in a more special setting.

TRAINING Several of the speakers were young scientists. During the lectures there were much more discussions and questions, than what I really had expected. Maybe the active atmosphere was the result of a favourable constellation of individuals, but the very pleasant site of San Feliu undoubtedly contributed also. There were also lots of discussion during the breaks, often involving both older and younger scientists. I regret that two grantees did not show up, in one case at least because of the strike at Air France on the arrival day. I believe that this meeting has been profitable for many of the young participants.

ADDITIONAL INFORMATION Acknowledgements: This conference was possible to organize

thanks to support from the European Commission Euroconference activity of the Training & Mobility of Researchers Programme and the European Science Foundation.

Palaiseau, 28.9.2000 Johannes Sjoestrand Centre de Mathematiques, Ecole Polytechnique Palaiseau Cedex, France

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ASTROgeomanal980439/EW/jb 1) BISMUT Jean-Michel 2) SJOESTRAND Johannes 1) Université Paris-Sud, Département de Mathématique, Bâtiment 425, 91405 Orsay, France 2) Ecole Polytechnique, Centre de Mathématiques, Département de Mathématiques, 91128 Palaiseau cedex, France 1) (33) 169 15 79 97 2) (33) 169 33 37 86 1) (33) 169 33 30 19 2) (49) 941 943 2576 1) [email protected] 2) [email protected] HENDEKOVIC Josip EURESCO Conferences, European Science Foundation, 1 quai Lezay-Marnésia, 67080 Strasbourg cedex, France Tel. +33 (0)388 76 71 35 Fax. +33 (0) 388 36 69 87 E-Mail: [email protected] ERBFMMACT 980439

Publications:

It is certainly premature to mention publications resulting from the event. However I have no doubt that many of the presentations were highly original, and some presented for the first time to an international audience.

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http://www.esf.org/euresco/99/PC99127a.htm http://www.esf.org/euresco/00/pc00127a.htm

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PROGRAMME OF EVENTS Event n°1: Analysis and geometry

Event n°2: analysis and spectralTheory

Dates: 04/06/99 - 09/06/99 5 day

Dates: 22/09/00 - 27/09/00 5 day

Place: Obernai (nr. Strasbourg), France

Place: San feliu de guixols, spain

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