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Summary: Pseudo differential operators, dirac operators, & diffusion
Review: In this second volume, of a set of three, the author takes up linear equations such as Dirac operators and those found in Brownian / diffusive processes. The first chapter presents fundamental results concerning pseudo-differential operators including Egorov's theorem and Weyl Calculus, while the second chapter presents spectral theory with applications to the laplace operator on n spheres & noncompact manifolds (such has hyperbolae and cones). A chapter on scattering from obstacles includes general discussion of wave operators, scattering from a sphere and the inverse scattering problem. Index theory, clifford algebra, spin manifolds, Chern -Gauss -Bonnet, and the Riemann -Roch theorem are covered in the chapter on dirac operators. The chapter on Brownian motion includes general discussion of Ito integrals and SDE as well as the Feynman-Kac solution of the heat eqaution. Finally, the Neumann boundary problem is the subject of the final chapter.As with the first volume of this set, the writing is dense and challenging. A reader should be a professional or advanced graduate student with access to a large library and plenty of time. However, if one wishes to have a definitive text that covers the modern understanding of PDE in general these three texts are it.
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Summary: Modern approach to important topics of mathematical physics.
Review: This second volume focuses on several interesting problems arising from mathematical physiscs, making a perfect connection between the purely theoretical digression and the sources from which it originated. The contents are: Pseudodifferential operators; spectral theory; scattering by obstacles; Dirac operators and index theory; Brownian motion and potential theory; the "D"-bar Neumann problem. Appendix: Connections and curvature. Useful for mathematics and physics advanced undergraduates, graduate students, and researchers. Plenty of excercises and references.
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