By Jean-Claude Zambrini

In 2003 the Xiv overseas Congress on Mathematical Physics (Icmp) was once held in Lisbon with greater than 500 contributors. Twelve plenary talks got in quite a few fields of Mathematical Physics: E Carlen «On the relation among the grasp equation and the Boltzmann Equation in Kinetic concept» A Chenciner «Symmetries and “simple” ideas of the classical n-body challenge» M J Esteban «Relativistic versions in atomic and molecular physics» ok Fredenhagen «Locally covariant quantum box conception» okay Gawedzki «Simple versions of turbulent delivery» I Krichever «Algebraic as opposed to Liouville integrability of the soliton platforms» R V Moody «Long-range order and diffraction in mathematical quasicrystals» S Smirnov «Critical percolation and conformal invariance» J P Solovej «The strength of charged topic» V Schomerus «Strings throughout the microscope» C Villani «Entropy creation and convergence to equilibrium for the Boltzmann equation» D Voiculescu «Aspects of loose probability». The e-book collects to boot rigorously chosen invited consultation Talks in: Dynamical platforms, Integrable structures and Random Matrix concept, Condensed topic Physics, Equilibrium Statistical Mechanics, Quantum box conception, Operator Algebras and Quantum info, String and M thought, Fluid Dynamics and Nonlinear Pde, common Relativity, Nonequilibrium Statistical Mechanics, Quantum Mechanics and Spectral thought, course Integrals and Stochastic research.

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Then for n large enough, ip^ is in //"1(M3, C 4 ) and there exists a solution of the HartreeFock equations, $ = (** £HF(*). 1 are shown to converge towards ground state solutions of the HartreeFock problem when c goes to +oo. This gives a "particular" status to that solution of the Dirac-Fock equations. It does not minimize the Dirac-Fock energy among all functions in the set £, since infs £c = — oo, but when taking the nonrelativistic limit, they approach those solutions of the Hartree-Fock equations which minimize the corresponding Hartree-Fock energy. **

2. N e w "simple" solutions: arbitrary masses I discuss two classes: in the first one, purely spatial, the homographic solutions — necessarily planar — are excluded because their action has a non positive Hessian, which forbids them to be even a local minimizer; in the second one, planar, they are excluded by the combination of symmetry constraints and a homotopy constraint. 1. Generalized HipHops: the simplest non planar solutions It follows immediately from Marchal's theorem that a minimizer of the action among loops of configurations with the Italian symmetry is collision-free.

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