Arithmetic Noncommutative Geometry

摘要奖：DMS-1007207首席研究员：Matilde Marcoll这项建议旨在利用非对易几何的技术，特别是基于泛函分析、算子代数、整体分析和指数理论的分析技术，来探讨与算术几何相关的问题。这一建议所涵盖的主要主题包括：与实二次域的实乘并行的非交换环面理论到具有实乘的椭圆曲线理论；使用与数域和函数域相关的量子统计力学系统作为显式类域理论的方法；应用泛函分析和量子统计力学的技术来研究纠错码的渐近问题；根据指数定理和谱三元组，从非交换几何构造具有p元均匀的曲线的不变量；研究动机与非交换空间之间的关系，与种类和动机的L函数的关系；将Riemann Zeta函数的非对易几何方法与算术拓扑、矩阵模型和分叶空间联系起来；了解模曲线非对易边界上的模性的表现。该项目范围是跨学科的，因为它的目的是探索非对易几何中的工具和技术之间的相互作用，并从数论和算术中激发问题。这提供了一种解决现代数学经典问题的新方法，我们计划使用源自量子力学、统计力学和量子场论的技术和思想。非对易几何特别适合作为一种工具，因为它恰好是一种适用于量子物理的几何学：因此，它提供了一种将量子力学的思想和技术与更经典的代数几何和数论方法相结合的方法。我们期望对这一方法的持续研究将通过微调现有理论以适应特定的数论问题，从而在非对易几何本身领域内带来进一步的进步和发展，同时它将为数论相关问题提供新的可能的方法和不同的观点。这项提案有很强的教育成分，有六名研究生参与了该项目的各个部分。还将涉及一个国际联系和合作网络。

AbstractAward: DMS-1007207Principal Investigator: Matilde MarcolliThis proposal aims at using techniques from noncommutative geometry, especially analytic techniques based on functional analysis, operator algebras, global analysis and index theory, to approach questions of relevance to arithmetic geometry. The main themes covered in this proposal include: the theory of noncommutative tori with real multiplication as a parallel for real quadratic fields to the theory of elliptic curves with real multiplication; the use of quantum statistical mechanical systems associated to number fields and function fields as an approach to explicit class field theory; an application of techniques from functional analysis and quantum statistical mechanics to the study of the asymptotic problem for error correcting codes; the construction of invariants of curves with p-adic uniformization from noncommutative geometry, in terms of index theorems and spectral triples; investigating the relation between motives and noncommutative spaces, in relation to L-functions of varieties and motives; relating the noncommutative geometry approach to the Riemann zeta function to arithmetic topology, matrix models, and foliated spaces; understanding manifestations of modularity on the noncommutative boundary of modular curves.The project is interdisciplinary in scope, as it is aimed at exploring the interactions between tools and techniques from noncommutative geometry and motivating problems from number theory and arithmetic. This presents a novel approach to classical questions of modern mathematics, where we plan on using techniques and ideas derived from the world of quantum mechanics, statistical mechanics, and quantum field theory. Noncommutative geometry is especially suitable as a tool, since it was developed precisely a a geometry adapted to quantum physics: as such, it provides a way to combine quantum mechanical ideas and techniques with more classical algebro-geometric and number-theoretic methods. We expect that a continuing investigation of this approach will lead to further advances and developments within the field of noncommutative geometry itself, by the challenge of fine tuning the available theory to fit specific number theoretic problems, while at the same time it will provide new possible approaches and different viewpoints on questions of number theoretic relevance. This proposal has a strong educational components, with six graduate students involved in various parts of the project. A network of international contacts and collaborations will also be involved.