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One of the main ideas in organizing the Summer Institute of Cargese on "Bifurcation Phenomena in Mathematical Physics and Related Topics" was to bring together Physicists and Mathematicians working on the properties arising from the non linearity of the phenomena and of the models that are used for their description. Among these properties the existence of bifurcations is one of the most interesting, and we had a general survey of the mathematical tools used in this field. This survey was done by M. Crandall and P. Rabinowitz and the notes enclosed in these proceedings were written by E. Buzano a]ld C. Canuto. Another mathematical approach, using Morse Theory was given by J. Smoller reporting on a joint work with C. Conley. An example of a direct application was given by M. Ghil. For physicists the theory of bifurcation is closely related to critical phenomena and this was explained in a series of talks given by J.P. Eckmann, G. Baker and M. Fisher. Some related ideas can be found in the talk given by T. T. Wu , on a joint work with Barry Mc Coy on quantum field theory. The description of these phenomena leads to the use of Pade approximants (it is explained for instance in the lectures of J. Nuttall) and then to some problems in drop hot moment problems. (cf. the lecture of D. Bessis).
List of contents
I Bifurcation in mathematics and applications.- Mathematical Theory of Bifurcation.- Remarks on the Stability of Steady-State Solutions of Reaction-Diffusion Equations.- Successive Bifurcations and the Ice-Age Problem.- II Bifurcation in Theoretical Physics and Statistical Mechanics; Applications.- Critical Phenomena in Statistical Mechanics Aspects of Renormalization Group Theory.- Lattice Renormalization of Non-Perturbative Quantum Field Theory.- The Continuous-Spin, Ising Model of Field Theory and the Renormalization Group.- Bifurcations for Maps.- Critical Properties of One Dimensional Mappings.- Intermittency: A Simple Mechanism of Continuous Transition from Order to Chaos.- Metal-Insulator Transition in One-Dimensional Deformable Lattices.- Sets of Minimum Capacity, Padé Approximants and the Bubble Problem.- Bifurcations in the Diophant Moment Problem.- III Yang-Mills Field Theory.- Yang-Mills Fields: Semi Classical Aspects.- Propagation of the Energy of Yang-Mills Fields.- Euclidean Yang-Mills and Related Equations.- Existence of Stationary States in Nonlinear Scalar Field Equations.- IV Solitons in Physics and in Algebraic Geometry.- Solitons: Inverse Scattering Theory and its Applications.- New Results on the Nonlinear Boltzmann Equation.- Solvable Many-Body Problems and Related Mathematical Findings (and Conjectures).- Riemann Monodromy Problem, Isomonodromy Deformation Equations and Completely Integrable Systems.- Padé Approximation and the Riemann Monodromy Problem.- V Nonlinear Partial Differential Equations and Applications.- Nonlinear Evolution Equations of Schrödinger Type.- Asymptotic Behavior of Solutions of Hyperbolic Balance Laws.- Nonlinear Problems Arising in the Study of Nematic Liquid Crystals.- Some Properties of Functional Invariant Sets forNavier-Stokes Equations.- An Introduction to the Time-Dependent Hartree-Fock Theory in Nuclear Physics.- List of Participants.
