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.

这位研究人员(格罗斯)和他的同事(卡兹丹和萨默斯)正在研究数论、表示论和代数几何中的一系列问题。格罗斯将研究四元数实群上模形式的傅立叶系数(与N.Wallach)，Siegel风格的例外theta对应(与W.T.Gan)，以及模为素数的模形式的各种理论。他还将在中间维度研究下村品种的亚种，并与固定的伽罗瓦群(与G.萨文)一起研究动机。Kazhdan将致力于朗兰兹提升和单幂晶体理论、环空间上的倒叶理论以及代数积分理论。萨默斯将研究幂零轨道覆盖的表示，并试图解决复李代数中幂零轨道闭合的正规性问题。他还将研究与复李群的么正表示之间的联系。该提案涉及数论、表示论和代数几何等数学分支领域中的几个问题。这些问题中的许多都是基于这样一种哲学，即代数信息可以通过几何方法获得。这项工作的中心是对称群或代数群的使用，它是一个本质上既是代数又是几何的对象。这些对称基团在物理和化学中自然产生。要说这个提案中问题的解决有朝一日会影响密码学、理论物理和量子计算的研究，这并不过分雄心勃勃。