Encyclopedia of Mathematical Physics

Professor Francoise graduated in Mathematics and in Physics from Grenoble University, France in 1975. He is currently professor of Mathematics at the Universite Pierre et Marie Curie, Paris and a member of the Laboratoire Jacques-Louis Lions after having held a position of charge de recherches at CNRS. Professor Francoise regularly travels and lectures abroad. He spent one year at IMPA (Rio de Janeiro, Brazil) in 1981 and one year at U.C. Berkeley in 1984. He was associate professor at University of Arizona, Tucson in 1987. Professor Francoise has delivered several series of lectures in Milan, in Rome, at the Banach Centre (Warsaw), at CRM (Montreal) and other institutes. He received the prize "du Fay? from the Academie des Sciences de Paris in 1989. His scientific publications include over eighty articles published in international journals and contributions to several books. His scientific research activity focuses on small oscillations near equilibrium of Hamiltonian systems, singularity theory of functions and vector fields, normal forms and semi-classical analysis, integrable systems, bifurcation theory of dynamical systems, finiteness properties of singular projections of analytic sets, bursting oscillations, synchronization and phase locking of weakly coupled oscillators and isochronous systems. Dr. Gregory L. Naber received all three of his degrees in Mathematics from Carnegie-Mellon University and has since held positions in Pennsylvania, California, Hong Kong and Tennessee. His areas of interest include differential topology and geometry and, most particularly, their interaction with mathematical physics. It is this interaction, and the desire to make it more widely known, appreciated and utilized in the mathematical community that has motivated nearly all of his published work, as well as his involvement with the Encyclopedia of Mathematical Physics. Dr. Tsou Sheung Tsun obtained her B.Sc. in Hong Kong and her Doctorat esSciences in Geneva. She has held research fellowships at Wadham College, Oxford, and at the Mathematical Institute, Oxford, where she is now on the Faculty. Trained both as a mathematician and a physicist, Dr. Tsou has worked in gauge theory, string theory and particle physics. Recently she has concentrated on theoretical problems connected with the generation puzzle, neutrino oscillation and electric-magnetic duality. She is also active in the European Mathematical Society and European Women in Mathematics.