Algebraic Geometry and Representation Theory in Genus One

内文斯将发展新的方法，并将它们应用于代数和表示理论-正如在(可能是奇异的)亏格一曲线上的丛模的几何和非交换代数中所编码的那样。首先，内文斯将系统地建立非对易代数的一些基本对象、双仿射Hecke代数及其退化的几何实现。其次，内文斯将通过一些普遍存在的粒子系统、Calogero-Moser系统和Ruijsenaars-Schneider系统的相空间的几何(经典和量子)几何，为这些几何对象开发“分类调和分析”工具。第三，内文斯将应用这些工具来对这些代数的表示进行分类，并在他们的研究中探索新的几何和代数方向。调和分析在数学及其应用的许多领域是一个强大的工具。例如，经典的傅里叶变换将波分解成其纯频率，在理解许多具体的物理问题中发挥着核心作用。内文斯将开发几何环境中的调和分析的新方法。然后，他将应用这些强大的新工具来探索被称为双仿射Hecke代数的重要代数对象的丰富结构。

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.