LOCAL-GLOBAL INTERACTION IN NONCOMMUTATIVE GEOMETRY

这个项目的首要目标是将非对易空间的特征类的局部表示推广到由Connes和Pi引入的III型谱三元组所描述的更复杂的空间。由此产生的局部-整体理论将被应用于横向几何分离、模形式、Hecke算符和Rankin-Cohen括号，并用于改进粒子物理标准模型的数学公式。在这个过程中，关于叶理特征类的Hopf循环理论将推广到所有经典类型的横向几何。在不同的方向上，将边界的概念引入到谱三重方法中，并利用相对循环上同调的方法来发展有边界的非对易空间的特征类。在不同的科学分支中，局部和全局的概念形成了理论的两个明显不同但共存的方面，这两个方面往往通过局部-全局原理以一种有趣的方式联系在一起。本项目将为在非对易几何的背景下实现这一原则开发新的工具，非对易几何是受量子力学启发的一种现代几何变体，它为非对易算符值坐标喝彩。虽然这一特征排除了从头开始任何基于点的天真的空间概念化，但受量子力学中的玻尔对应原理的启发，对局域性概念有更微妙的解释，这一点将得到充分利用。这项拟议的工作将在数学和物理的几个领域之间产生新的联系，从而促进它们相互丰富的互动。

ABSTRACTThe foremost objective of this project is to extend the local representation of thecharacteristic classes of noncommutative spaces to the more intricate spaces described by the spectral triples of type III introduced by Connes and the PI. The resulting local-global theory will be applied to the transverse geometry offoliations, to modular forms, Hecke operators and Rankin-Cohen brackets, and to the refinement of the mathematical formulation of the standard model of particle physics. In the process, the Hopf cyclic theory of characteristic classes for foliations will be extended to all classical types of transverse geometries. In a different direction, the concept of boundary will be incorporated in the spectral triple approach, and the characteristic classes of the noncommutative spaces with boundary will be developed by means of relative cyclic cohomology. In various branches of science, the concepts of local and global form two markedly different but coexisting facets of a theory, which are often correlated in an interesting way by a local-global principle. The present project will develop newtools for the implementation of this principle in the setting of noncommutative geometry, a modern variant of geometry inspired by quantum mechanics whichallows for non-commuting operator-valued coordinates. Although this feature precludes ab initio any naive spatial conceptualization based on points, there is a more subtle interpretation of the notion of locality, inspired by the Bohr correspondence principle in quantum mechanics, which will be fully exploited. The proposed work will engender new connections between several fields of mathematics and physics, thus stimulating their mutually enriching interaction.