Mathematical Sciences: Dynamical Methods in Variational Problems

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Loss and Carlen will continue their investigations into problems in nonlinear analysis which have their origins in physics. One of the main efforts will be to obtain an extension of a new approach for minimizing nonlinear functionals, called competing symmetries. These methods will be applied to harmonic analysis, physics and geometry. Along similar lines, they will study the approach to equilibrium in the Boltzmann equations by means of entropy production inequalities. 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 concepts 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 theory.