Mathematical Sciences: Spectra and Geometry: Random Media and Hyperbolic Manifolds

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Hislop will continue his study of the interplay of effective geometry and spectral properties. The first part of the project concerns spectral and transport properties of random media. He will investigate Anderson localization for a family of continuous random Hamiltonians and for Maxwell's equations in disordered media. The second part of the project will concentrate on infinite volume hyperbolic manifolds. He will complete the characterization of the Eisenstein series and S-matrix for geometrically finite hyperbolic manifolds. 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.