Algebraic Geometry and Representation Theory in Genus One

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Nevins will develop new methods and apply them to algebra and representation theory ``in genus one''---that is, as encoded in the geometry and noncommutative algebra of moduli of bundles on (possibly singular) genus one curves. First, Nevins will systematically establish geometric realizations of some fundamental objects of noncommutative algebra, the double affine Hecke algebras and their degenerations. Second, Nevins will develop tools of ``categorical harmonic analysis'' for these geometric objects via the geometry (classical and quantum) of phase spaces of some ubiquitous particle systems, the Calogero-Moser and Ruijsenaars-Schneider systems. Third, Nevins will apply these tools to classify representations of these algebras and explore new geometric and algebraic directions in their study.Harmonic analysis is a powerful tool in many areas of mathematics and its applications. For example, the classical Fourier transform, which decomposes a wave into its pure frequencies, plays a central role in the understanding of many concrete physical problems. Nevins will develop new methods of harmonic analysis in a geometric setting. He will then apply these powerful new tools to explore the rich structure of important algebraic objects known as double affine Hecke algebras.