Research in Noncommutative and Transverse Geometry

非交换几何和横向几何研究

• 批准号: 0245481
• 负责人: Henri Moscovici
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245481&HistoricalAwards=false
• 关键词: Research; Noncommutative; Transverse; Geometry

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.

莫斯科维奇本项目将继续几年前由首席研究员与阿兰·康纳斯合作发起的项目，为理解横向几何提供一个一般的环境。一个主要的新目标包括在应用程序的背景下，模块化的形式和模块化的对应关系，以揭示隐藏的兼容性之间的逐点积的模块化形式和行动的Hecke运营商。 拟议的研究的另一个主要目标是阐述的K-理论解释的Godbillon-Vey类和其他二级不变量，通过细化的运营商理论的本地指数公式，位于该计划的核心。拟议的工作将有助于发展新的数学工具，用于处理大量的非经典空间，这些空间具有可辨别的几何意义，但不能通过交换坐标进行充分描述。 这种空间，出现在数学和物理的许多领域，有一个共同的特征，即它们的可观测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延伸和反思，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非经典对称性原理。