Geometric structures over the absolute point and their arithmetic

具有相关算术内容的“绝对”几何理论的发展代表了基础研究的一个新方向，它将代数几何、非对易几何、数论和数学物理等领域联系起来。正特征全局场结构的代数曲线几何，在解决数论中几个基本问题的过程中，已经显示出它的关键作用，这些问题对于特征为零的全局场来说，仍然是开放的。一些组合公式，如在基数为q的有限域上提供格拉斯曼有理点集的基数的方程，当q=1时也是保持有意义值的有理表达式。A. Weil和K. Iwasawa认为，相邻的单位根是一个类似于基域的扩展的构造过程，也激发了对一个数学对象的搜索，该数学对象被期望为伽罗瓦域的非平凡极限，因为这些域的基数（q）趋于1。在Hasse-Weil zeta函数中取极限q为1的过程已经被证明可以确定“绝对曲线”的“计数函数”N（q），在这个基本框架中，它是黎曼zeta函数的几何对应物。此外，委员会认为，基于Bost-Connes量子统计力学系统（它最初给出了非交换几何和数论之间的关系）和相应的几何空间（即有理数的整体域的adele类空间）计数函数N（q）被解释为与Riemann-Weil显式公式和L的零点的谱解释密切相关的交集数。拟研究的目标是揭示几何的adele类空间及其变种，并比较这些空间与“绝对曲线”，使用的工具，代数几何，非交换几何，数论（包括岩泽理论），和热带几何。拟议的项目是致力于塑造建设和促进一个新的几何结构的基本性质的研究目的是移植A.韦伊在数论，在他的证明黎曼假设的功能领域，以案件的代数数域。所采用的方法主要来自代数几何、非交换几何、数论（包括岩泽理论）和热带几何，计算机测试是重要工具。该项目的广泛影响是向年轻研究人员和资深科学家传达需要通过努力发展数论和非对易几何领域之间的新联系来攻击算术中一些主要开放问题的概念理解，非对易几何虽然是一个非常新的数学领域，但在最近几年中迅速成熟。这项研究解决了涉及数论中一些关键数学对象的中心问题和开放问题，例如着名的Riemann zeta函数，其推广（L-函数）和岩泽理论中的p-adic对应物。该项目的发展也有望将非对易几何、数论和“特征一”数学（包括热带几何）之间的相互作用统一在一个统一的方法论之下。

The development of an "absolute" geometric theory with relevant arithmetic contents represents a new direction of fundamental research, interconnecting the fields of algebraic geometry, noncommutative geometry, number-theory and mathematical physics. The geometry of algebraic curves underlying the structure of global fields of positive characteristic has shown its crucial role in the process of solving several fundamental questions in number-theory which are still open for global fields of characteristic zero. Some combinatorial formulas, like the equation supplying the cardinality of the set of rational points of a Grassmannian over a finite field of cardinality q are known to be rational expressions keeping a meaningful value also when q=1. The classical point of view of A. Weil and K. Iwasawa that adjoining roots of unity is a process analogous to the construction of an extension of a base field, also motivates the search of a mathematical object that is expected to be a non-trivial limit of Galois fields as the cardinality (q) of these fields tends to 1. The process of taking the limit q to 1 in the Hasse-Weil zeta function has been shown to determine the "counting function" N(q) of the "absolute curve" which is the geometric counterpart, in this basic framework, of the Riemann zeta function. Moreover, based on the Bost-Connes quantum statistical mechanical system (which originally gave the relation between noncommutative geometry and number theory) and on the corresponding geometric space (i.e. the adele class space of the global field of rational numbers) the counting function N(q) is interpreted as an intersection number closely related to both the Riemann-Weil explicit formulas and the spectral interpretation of the zeros of L-functions.The goal of the proposed research is to unveil the geometry of the adele class space and its variants, and to compare these spaces with the "absolute curve", using the tools of algebraic geometry, noncommutative geometry, number theory (including Iwasawa theory), and tropical geometry.The proposed project is devoted to shape the construction and promote the study of the fundamental properties of a new geometric structure with the goal to transplant the ideas of A. Weil in number-theory, in his proof of the Riemann Hypothesis for function fields, to the case of algebraic number fields. The methods to be used come mainly from algebraic geometry, noncommutative geometry, number theory (including Iwasawa theory) and tropical geometry, with computer testing as an important tool. The broad impact of the project is to communicate to an audience of young researchers and senior scientists the need to attack the conceptual understanding of some of the main open problems in arithmetic by working on the development of new connections between the fields of number theory and noncommutative geometry which although a very new area of mathematics, has matured rapidly in the recent past few years. This research addresses central aspects and open problems that involve some of the key mathematical objects in number-theory, such as the celebrated Riemann zeta function, its generalizations (L-functions) and the p-adic counterparts in Iwasawa theory. The development of this project is also expected to solidify the interactions between noncommutative geometry, number theory and mathematics in \characteristic one", including tropical geometry, under a unified methodology.