Preface The XIVth International Congress on Mathematical Physics (ICMP) was held in Lisbon, Portugal, from July 28 to August 2, 2003, at the Aula Magna of the University of Lisbon. For two days preceding the Congress, a Young Researchers Symposium (YRS), aimed mainly at the younger participants in ICMP, was held at the Instituto Superior Tecnico. Most of the invited talks at the YRS, including two of the three Poincare Prize Lectures (awarded during the ICMP), by Huzihiro Araki and Elliott Lieb, are collected here. They provide a stimulating overview of current prospects in Mathematical Physics. It was still possible, 25 years ago, to do excellent mathematics without the slightest interest for theoretical physics, but nowadays few active mathematicians would defend this attitude. As the main historical source of mathematical ideas, theoretical physics and the innumerable challenges it provides are currently re- garded as part of the cultural background of mathematics. We hope that the contributions included in this volume will illustrate this evolution, which may well have worked in favour of the present Golden Age of Mathematics. The book of ICMP 2003 was published in 2005 by World Scientific. We take this new opportunity to thank our sponsors and to list them: International Association of Mathematical Physics European Commission, High-Level Scientific Conferences, contract HPCF-2002-00368 International Union of Pure and Applied Physics In- ternational Mathematical Union Clay Mathematics Institute Portuguese Founda- tion for Science and Technology Calouste Gulbenkian Foundation Daniel Iagol- nitzer Foundation European Physical Society Abdus Salam International Centre for Theoretical Physics National Science Foundation USA French Embassy in Por- tugal Portuguese Mathematical Society Luso-American Foundation Portuguese Ministry of Education Oriente Foundation Nestle Portugal Lisbon's City Hall Rectorship of the University of Lisbon Group of Mathematical Physics of the University of Lisbon Center for Mathematical Analysis, Geometry, and Dynamical Systems Center for Mathematics and its Applications Center for Mathematics and Fundamental Applications Center for Linear Structures and Combinatorics Amer- ican Institute of Physics Birkhiiuser-Verlag Cambridge University Press Institute of Physics Kluwer Academic Publishers Oxford University Press Springer-Verlag World Scientific Publishers. The Editors vii

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