FRG Collaborative Research: Noncommutative Geometry and Number Theory

FRG 合作研究：非交换几何与数论

• 批准号: 0652431
• 负责人: Caterina Consani
• 金额: $ 21万
• 依托单位: Johns Hopkins 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 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652431&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.

非对易几何与数论的相互作用代表了一个新的方向，在过去的几年里迅速成熟。该合作研究项目致力于将非对易几何的方法和工具应用于数论中的特定主题，与研究显式类场理论问题(希尔伯特第12问题)、Riemann Zeta函数和代数族的L函数有关。一个预期的结果将是在循环上同调的背景下将Weilexplex公式理解为Lefschetz迹公式。该项目的另一个中心方面涉及用最近对Q格模可公度的非交换空间的量子统计力学性质的研究得出的想法来补充马宁的方法来补充斯塔克关于实二次域的猜想(通过实乘的非对易环面)。从横向几何概念和结构到模形式的设置的转移，有望得到关于模形式和Hecke算子的新结果。我们将利用非对易几何中的谱三元组的形式和局部折射率公式来研究比芒福德曲线更一般的刚性解析空间。在揭示量子场论中费曼图的剩余与混合状态主题周期之间的关系方面也有望取得重大进展。这一合作研究项目旨在阐明与非对易几何、数论和数学物理领域之间丰富的和大部分不稳定的相互联系有关的一些重要主题。这些主题涉及后一领域中的一些关键数学对象，例如著名的黎曼Zetaftion函数及其在数论中称为L函数的推广，以及在微扰量子场论中的费曼积分。他们的研究将以一种新颖和统一的方式进行，通过非对易几何的方法，这是一门由数学中最古老的分支之一几何和最年轻的量子力学融合而成的学科。