Algebraic Geometry and Representation Theory in Genus One

属一中的代数几何与表示论

• 批准号: 0757987
• 负责人: Thomas Nevins
• 金额: $ 12万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0757987&HistoricalAwards=false
• 关键词: Algebraic; Geometry; Representation; Theory; Genus

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.

Nevins将开发新的方法，并将其应用于代数和“in genus one”的表示理论——也就是说，编码在（可能是奇异的）genus one曲线上束模的几何和非交换代数中。首先，内文斯将系统地建立非交换代数、双仿射Hecke代数及其退化的一些基本对象的几何实现。其次，Nevins将通过一些普遍存在的粒子系统（Calogero-Moser和rujsenaars - schneider系统）的相空间的几何（经典和量子），为这些几何物体开发“绝对谐波分析”工具。第三，Nevins将应用这些工具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。谐波分析在许多数学领域及其应用中是一个强有力的工具。例如，经典的傅立叶变换，将一个波分解成它的纯频率，在理解许多具体的物理问题中起着核心作用。内文斯将在几何环境中发展谐波分析的新方法。然后，他将应用这些强大的新工具来探索被称为双仿射赫克代数的重要代数对象的丰富结构。