Algebra, Number Theory and Algebraic Geometry

代数、数论和代数几何

• 批准号: 0070674
• 负责人: Benedict Gross
• 金额: $ 52.7万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070674&HistoricalAwards=false
• 关键词: Algebra; Number; Theory; Algebraic; Geometry; Algebra; Number; Theory; Algebraic; Geometry

The investigator (Gross) and his colleagues (Kazhdan and Sommers) areworking on an assortment of problems in number theory, representationtheory, and algebraic geometry. Gross will study Fourier coefficients formodular forms on quaternionic real groups (with N. Wallach), theexceptional theta correspondences in the style of Siegel (with W.T. Gan),and various theories of modular forms modulo a prime. He will alsoinvestigate subvarieties of Shimura varieties in the middle dimension andmotives with a fixed Galois group (with G. Savin). Kazhdan will work onLanglands' lifting and the theory of unipotent crystals, perverse sheaveson loop spaces, and the theory of algebraic integration. Sommers willstudy representations arising from covers of nilpotent orbits and attemptto resolve the normality question for the closure of nilpotent orbits in acomplex Lie algebra. He will also study connections with unitaryrepresentations of complex Lie groups. The proposal deals with several questions in the subfields ofmathematics known as number theory, representation theory, and algebraicgeometry. Many of these questions are motivated by the philosophy thatalgebraic information can be obtained by geometric methods. At the centerof the work is the use of a symmetry group, or algebraic group, which isan object that is both algebraic and geometric in nature. These symmetrygroups arise naturally in physics and chemistry. It is not too ambitiousto say that the solution to the problems in this proposal will one dayaffect research in cryptography, theoretical physics, and quantumcomputing.

研究人员（Gross）和他的同事（Kazhdan和Sommers）正在研究数字理论，代表理论和代数几何形状中的各种问题。 Gross将在Quaternionic Real群体（带有N. Wallach），Siegel（带有W.T. GAN）风格的TheTa theta对应方面研究傅立叶系数形式以及模块化形式的各种理论。 他还将通过固定的Galois组（与G. Savin）进行中等维度和动力的Shimura品种的亚村庄进行评估。 Kazhdan将在Onlanglands的提升和一般晶体，不正当的Sheaveson Loop空间和代数整合理论的理论上工作。 索默斯（Sommers）威尔（Sommers Will）的表示是由nilpotent轨道的覆盖物引起的，并试图解决正态性问题，以关闭Acomplex中的nilpotent Orbits lie代数。他还将研究复杂谎言群体的统一分数的联系。该提案涉及数字理论，表示理论和代数测定法的数字子领域的几个问题。 这些问题中的许多是出于哲学的动机，即可以通过几何方法获得algebraic信息。 工作的中心是使用对称组或代数群的使用，该对象本质上是代数和几何的Isan对象。 这些对称组自然出现在物理和化学中。不太容易说，解决该提案中问题的解决方案将在密码学，理论物理学和量子量表中进行日间研究。