FRG Collaborative Research: Noncommutative Geometry and Number Theory

FRG 合作研究：非交换几何与数论

• 批准号: 0651925
• 负责人: Matilde Marcolli
• 金额: $ 4.5万
• 依托单位: Florida State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0651925&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Noncommutative; Geometry

The interaction between noncommutative geometry and number theoryrepresents a new direction, which has rapidly matured in the pastfew years. The proposed collaborative research project is devoted toapplying the methods and tools of noncommutative geometry tospecific topics in number theory, pertaining to the study of theexplicit class field theory problem (Hilbert's 12th problem), of theRiemann zeta function and of the L-functions of algebraic varieties.One anticipated outcome will be a novel understanding of the Weilexplicit formulae as Lefschetz trace formulae in the context ofcyclic cohomology. Another central aspect of the project involvessupplementing Manin's approach to Stark's conjectures for realquadratic fields (via noncommutative tori with real multiplication)with ideas stemming from the recent investigation of the quantumstatistical mechanical properties of noncommutative spaces ofQ-lattices modulo commensurability. New results on modular forms andHecke operators are expected, arising from the transfer oftransverse geometry concepts and constructions to the setting ofmodular forms. The formalism of spectral triples together with thelocal index formula in noncommutative geometry will be exploited toinvestigate rigid analytic spaces more general than Mumford curves.Significant progress is also anticipated in the uncovering of therelationship between residues of Feynman graphs in quantum fieldtheory and periods of mixed Tate motives.This collaborative research project aims to shed light on a numberof important topics pertaining to the rich and largely untappedinterconnection between the fields of noncommutative geometry,number theory and mathematical physics. These topics address centralaspects and open problems, that involve some of the key mathematicalobjects in the latter fields, such as the celebrated Riemann zetafunction and its generalizations called L-functions in number theoryand Feynman integrals in perturbative quantum field theory. Theirinvestigation will be approached in a novel and unified manner,through the methods of noncommutative geometry, a discipline whichgrew out of the fusion between one of the oldest branches ofmathematics -- geometry, and one of the youngest -- quantummechanics.

非交通性几何形状与数字理论代表者之间的相互作用是一个新的方向，在过去的几年中已经迅速成熟。拟议的协作研究项目致力于旨在介绍数字理论中非共同的几何学上的主题的方法和工具，这与对义务阶级域理论问题的研究有关（希尔伯特的第十二个问题），theriemann zeta zeta zeta功能和对综合的综合征材的理解。痕量公式在环境的辅助学背景下。 该项目的另一个核心方面涉及Manin对Stark对实体领域的猜想的方法（通过非交通性的托里（通过具有真实乘法））的想法，其想法是由于最近研究的量子力学机械性能的不相同空间OFQ-Lattices Ofq-Lattices Modulo的量子能力。预期的是，对模块化形式的新结果和构造的转移概念和结构的转移到了模块化形式的设置。 The formalism of spectral triples together with thelocal index formula in noncommutative geometry will be exploited toinvestigate rigid analytic spaces more general than Mumford curves.Significant progress is also anticipated in the uncovering of therelationship between residues of Feynman graphs in quantum fieldtheory and periods of mixed Tate motives.This collaborative research project aims to shed light on a numberof important topics与非交通性几何，数理论和数学物理学领域之间的丰富且在很大程度上不开发的结合有关。这些主题涉及涉及后者领域中一些关键数学对象的中心问题，例如著名的Riemann Zetafunction及其在扰动量子域中的数字理论和FEYNMAN积分中称为L-功能的概括。通过非交通性几何学方法，将以新颖和统一的方式进行他们的研究，这是一种纪律，该学科脱离了最古老的大型神经学分支之一 - 几何形状和最年轻的量化 - 量化 - 量化 - QuantMechanics之一。