FRG Collaborative Research: Noncommutative Geometry and Number Theory

非对易几何与数论的相互作用代表了一个新的发展方向，在过去的几年中迅速成熟起来。拟进行的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课题，涉及显式类场论问题（希尔伯特第12问题）、黎曼zeta函数和代数簇的L-函数的研究，其中一个预期成果将是在循环上同调的背景下对Weil显式公式作为Lefschetz迹公式的新理解。 该项目的另一个核心方面涉及补充马宁的方法斯塔克的代数realquadratic领域（通过非交换环面与真实的乘法）的想法源于最近的调查quantumstatistical力学性质的非交换空间的Q-格模可积性。从横截几何的概念和结构转移到模形式的设置上，期望得到模形式和Hecke算子的新结果。谱三元组的形式和非对易几何中的局部指数公式将被用来研究比Mumford曲线更一般的刚性解析空间。在揭示量子场论中Feynman图的留数和混合Tate动机周期之间的关系方面也有望取得重大进展。这个合作研究项目旨在阐明与丰富和广泛的量子场论相关的一些重要课题非对易几何、数论和数学物理领域之间的未触及的联系。这些主题解决centralaspects和开放的问题，涉及到一些关键的programmalobject在后者的领域，如著名的黎曼zeta函数和它的推广称为L-函数在数论和费曼积分微扰量子场论。他们的调查将在一个新的和统一的方式，通过非对易几何的方法，一个学科whichgrow之间的融合最古老的数学分支之一-几何，和最年轻的-量子力学。

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.