9401777 Soffer Two main topics will be investigated. In the Cauchy problem for the nonlinear wave equation, the results of Ginibre-Soffer- Velo on the critical power nonlinear wave equation which are complete for the radial case will be further developed to include the general data. It is based on applying new Lebesgue pth power bounds which allow the control of the p-norm of a function in terms of singular weighted norms and partial regularity. In the theory of three body dispersive systems a new class of dilations, deformed by various partitions of unity to cluster decompositions will be used. This will allow proofs of local decay and other spectral properties of three body dispersive equations. Modern physics, quantum mechanics and relativity, is a product of the twentieth century. It is founded firmly in the last century's attempt to address the microstructure of matter and to come to grips with the concept of action-at-a distance, electro-magnetism, and heat radiation. The mathematical foundations for these developments collectively called mathematical physics, ranges from detailed analysis of Schroedinger operators, which governs the dynamics of particles, to unified field theory, which attempts to unite the four known forces into a single force. ***
