Arithmetic Noncommutative Geometry

摘要：项目编号：dms -1007207项目负责人：Matilde marcollii本项目旨在利用非交换几何中的技术，特别是基于泛函分析、算子代数、全局分析和指标理论的解析技术，来解决与算术几何相关的问题。本文的主题包括：实乘的非交换环面理论与实乘椭圆曲线理论的平行；使用与数场和函数场相关的量子统计力学系统作为显式类场论的一种方法；泛函分析和量子统计力学技术在纠错码渐近问题研究中的应用用指数定理和谱三元组构造非交换几何p进均匀化曲线的不变量研究动机与非交换空间之间的关系，涉及到变量和动机的l函数；将黎曼ζ函数的非交换几何方法与算术拓扑、矩阵模型和叶状空间联系起来；模曲线非交换边界上的模性表现。该项目在范围上是跨学科的，因为它旨在探索非交换几何中的工具和技术之间的相互作用，以及数论和算术中的激励问题。这提出了一种新的方法来解决现代数学的经典问题，我们计划使用来自量子力学、统计力学和量子场论的技术和思想。非交换几何特别适合作为一种工具，因为它是一种适合量子物理的精确几何：因此，它提供了一种将量子力学思想和技术与更经典的代数几何和数论方法结合起来的方法。我们期望对这种方法的持续研究将导致非交换几何本身领域的进一步进步和发展，通过微调现有理论以适应特定数论问题的挑战，同时它将为数论相关问题提供新的可能方法和不同的观点。该提案具有很强的教育成分，六名研究生参与了项目的各个部分。还将建立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.