Geometry and Physics - Noncommutative Geometry

该奖项为几何和物理系列暑期学校(GAP)的两个版本提供部分支持。Gap VIII(2010)将专注于非对易几何的微分几何方面，而Gap IX(2011)将致力于非对易几何和数学物理的代数几何方面。非对易几何旨在研究一类新的“空间”的几何性质，这些“空间”的函数代数不再是交换的。核心思想回到了量子力学，在量子力学中，位置和动量等可观测变量是不可交换的。由于它与数学和数学物理的多个领域密切相关，包括分层理论、指数理论、表示理论、代数几何、量子场论、量子化理论和弦理论，近年来这一课题得到了极大的探索。该项目包括两个暑期班(GAP VIII和GAP XI)，旨在收集非对易几何的不同流派和实例，以及确定新的新兴方向。拟议活动的主要目的是将从事迅速发展的学科的主要专家和青年研究人员聚集在一起，促进数学家和物理学家以及从事相关领域不同方面工作的团体之间的互动，以促进互动，鼓励不同领域之间的交流，并促进传播当前研究的最新成果。有许多有才华的年轻人在这个地区工作。GAP八和GAP十一将为他们提供传播他们的想法和拓宽他们的视野的绝佳机会。PIS预计将邀请大量博士后和研究生级别的数学家(来自欧洲、美国和发展中国家)，并举行海报或短谈会议，年轻的研究人员将有机会介绍他们的工作。这些学校将为年轻的美国科学家提供一个与欧洲同行交流想法和开展合作的绝佳机会。该项目由几何分析程序、代数和数论程序以及分析程序共同资助。

This award provides partial support for two editions of a series of summer schools on Geometry and Physics (GAP). GAP VIII (2010) will concentrate on differential geometric aspects of noncommutative geometry, while GAP IX (2011) will be devoted to algebraic geometric aspects of noncommutative geometry and mathematical physics. Noncommutative geometry aims to study geometric properties of a new class of "spaces" whose algebras of functions are no longer commutative. The central idea goes back to quantum mechanics, in which observables such as position and momentum do not commute. The subject has undergone tremendous exploration recently due to its close connection to a number of areas of mathematics and mathematical physics, including foliation theory, index theory, representation theory, algebraic geometry, quantum field theory, quantization theory, and string theory. The project consists in a pair of summer schools (GAP VIII and GAP XI) aimed at gathering different streams and instances of noncommutative geometry, as well as identifying new emerging directions. The main purposes of the proposed activities are to bring together leading experts and young researchers working in rapidly developing subjects, and to promote interaction between mathematicians and physicists, and groups working on different aspects of related areas so as to foster interaction, encourage cross-fertilization between different fields, and promote the dissemination of the most recent results of current research. There are many talented young people working in this area. GAP VIII and GAP XI will provide them with excellent opportunities to disseminate their ideas and to broaden their perspective. The PIs anticipate inviting a substantial number of mathematicians at the postdoctoral and graduate student level (from Europe, the United States, and developing countries), and holding poster or short-talk sessions through which young researchers will be given the chance to present their work. The schools will provide an excellent opportunity for young American scientists to exchange ideas with their European peers and initiate collaborations. This project is jointly funded by the Geometric Analysis Program, the Algebra and Number Theory Program, and the Analysis Program.