FRG Collaborative Research: Noncommutative Geometry and Number Theory

非交换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起来。提出的合作研究项目致力于将非交换几何的方法和工具应用于数论中的特定主题，涉及显式类场论问题（希尔伯特的第12个问题），黎曼ζ函数和代数变体的l函数的研究。一个预期的结果将是一个新的理解Weilexplicit公式作为Lefschetz迹公式在循环上同调的背景下。该项目的另一个核心方面涉及补充Manin的方法，以补充Stark对实二次域的猜想（通过实乘法的非交换环面），其思想源于最近对q格模可通约性的非交换空间的量子统计力学性质的研究。关于模形式和赫克算子的新结果是预期的，由于横向几何概念和结构转移到模形式的设置。谱三元组的形式化与非交换几何中的局部指标公式将被用来研究比Mumford曲线更一般的刚性解析空间。在揭示量子场论中费曼图的残数与混合泰特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一些重要的主题，这些主题与非交换几何、数论和数学物理领域之间丰富而尚未开发的相互联系有关。这些主题涉及中心方面和开放问题，涉及后一个领域的一些关键数学对象，例如数论中著名的黎曼ζ函数及其推广称为l函数，以及摄动量子场论中的费曼积分。他们的研究将通过非交换几何的方法以一种新颖而统一的方式进行，非交换几何是数学中最古老的分支之一——几何和最年轻的分支之一——量子力学融合而成的一门学科。

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.