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

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.

调查员（格罗斯）和他的同事（Kazhdan和Sommers）正在研究数论、表示论和代数几何中的各种问题。格罗斯将研究四元数真实的群（N。Wallach），Siegel（with W. T. Gan），以及模素数模形式的各种理论。 他还将研究中维的志村簇的子簇和具有固定伽罗瓦群的动机（G。Savin）。 Kazhdan将工作对朗兰兹'解除和理论的单能晶体，反常的shereton循环空间，和理论的代数积分。 Sommers将研究由幂零轨道覆盖引起的表示，并试图解决复杂李代数中幂零轨道闭包的正规性问题。他还将研究与复杂李群的酉表示的联系。该建议涉及数学的子领域，如数论，表示论和代数几何中的几个问题。 许多这些问题的动机是哲学，代数信息可以通过几何方法获得。 工作的中心是使用对称群或代数群，这是一个对象，既是代数和几何的性质。 这些基团在物理学和化学中自然产生。可以毫不夸张地说，这个方案中的问题的解决方案有一天会影响密码学、理论物理学和量子计算的研究。