Representation Theory and Harmonic Analysis on Homogeneous Spaces

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The proposed research covers a broad spectrum of problems and ideas reaching from abstract harmonic analysis on symmetric spaces, geometry, and infinite dimensional analogues, to classical questions in abelian harmonic analysis. The basic subject of the proposed research is harmonic analysis on symmetric spaces G/H which connects analysis, group theory (symmetry) and geometry. The project includes holomorphic extension of H-spherical distributions, applications of intertwining operators and evaluations of their spectrum, analysis on infinite dimensional spaces, representation theory and G-invariant Banach spaces of functions. The physical parts include reflection positivity and transfer operators. This work combines methods and ideas from several areas of mathematics: complex analysis, group actions on complex manifolds, classical harmonic analysis, applied mathematics, and quantum field theory.Harmonic analysis and representation theory are two central subjects of mathematics and play an important role in pure and applied mathematics as well as theoretical physics. Well known examples of applications are tomography and signal analysis, but the proposed research deals with more abstract and fundamental questions. The proposal is related to several parts of mathematics, both pure and applied, and includes several connections to physics, in particular quantum field theory in form of reflection positivity, and applied sciences. The research involves wide spectrum of collaboration including graduate students and a network of other researchers in the USA, Mexico, and Europe.