Geometry and Physics - Noncommutative Geometry

该奖项为两个版本的几何和物理系列暑期学校（GAP）提供部分支持。GAP VIII（2010）将专注于非交换几何的微分几何方面，而GAP IX（2011）将致力于非交换几何和数学物理的代数几何方面。非交换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可交换的。其中心思想可以追溯到量子力学，在量子力学中，位置和动量等可观测物是不能交换的。由于与叶理理论、指标理论、表示理论、代数几何、量子场论、量子化理论和弦理论等许多数学和数学物理领域的密切联系，这门学科最近经历了巨大的探索。该项目包括两个暑期学校（GAP VIII和GAP XI），旨在收集不同的非交换几何的流和实例，并确定新的新兴方向。所提议的活动的主要目的是将在快速发展的学科中工作的主要专家和年轻研究人员聚集在一起，促进数学家和物理学家之间的互动，以及在相关领域的不同方面工作的小组之间的互动，以促进互动，鼓励不同领域之间的交叉施肥，并促进当前研究的最新成果的传播。有许多有才华的年轻人在这个地区工作。GAP VIII和GAP XI将为他们提供极好的机会来传播他们的想法和扩大他们的视野。pi计划邀请相当数量的博士后和研究生水平的数学家（来自欧洲、美国和发展中国家），并举办海报或简短的谈话会议，通过这些会议，年轻的研究人员将有机会展示他们的工作。这些学校将为年轻的美国科学家提供与欧洲同行交流思想和开展合作的绝佳机会。本项目由几何分析项目、代数与数论项目和分析项目共同资助。

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.