Research in Noncommutative and Transverse Geometry

莫斯科这个项目将继续几年前由首席研究人员与阿兰·康尼斯合作发起的计划，为理解横向几何提供一个一般的环境。一个主要的新目标在于将该程序应用到模形式和模对应的上下文中，以揭示模形式的逐点乘积和Hecke算子的作用之间的隐藏兼容性。拟议研究的另一个主要目标是通过对位于程序核心的算子理论局部指数公式的改进，对Godillon-Vey类和其他次级不变量进行K理论解释。拟议的工作将有助于开发新的数学工具来处理大量非经典空间，这些空间具有明显的几何意义，但不能通过交换坐标的方式进行充分描述。这类空间产生于数学和物理的许多领域，它们有一个共同的特点，即它们的可观察到的局部参数表现得像无限维矩阵而不是数值变量。为了有效地处理它们，需要对已知的几何技术进行大量的拉伸和重新思考，这一过程导致了非交换几何的发展。本项目的一个显著特点是系统地利用了非经典对称原理。

AbstractMoscoviciThis project will continue the program initiated several years ago by the principal investigator in collaboration with Alain Connes, providing a general setting for the understanding of transverse geometry. A main new objective consists in the application of this program to the context of modular forms and modular correspondences, in order to uncover the hidden compatibility between the pointwise product of modular forms and the action of the Hecke operators. Another major goal of the proposed research is the elaboration of a K-theoretic interpretation for the Godbillon-Vey class and for other secondary invariants, by means of a refinement of the operator-theoretic local index formula that lies at the core of the program. The proposed work will contribute to the development of new mathematical tools for the treatment of a multitude of non-classical spaces, which have a discernible geometric meaning but cannot be adequately described by means of commuting coordinates. Such spaces, arising in many areas of mathematics and physics, have in common the feature that their observable local parameters behave like infinite-dimensional matrices rather than numerical variables. To deal with them efficiently much stretching and rethinking of the known geometric techniques is required, process that has led to the development of noncomutative geometry. A distinctive feature of the present project is the systematic utilization of non-classical symmetry principles.