Noncommutative Geometry at the Newton Institute

该奖项是对参加2006年7月24日至12月22日在英国剑桥牛顿研究所举行的为期一个学期的非交换几何研讨会的美国参与者的部分支持。非交换几何旨在将几何概念引入一类新的空间，这些空间的函数代数不再是可交换的。中心思想可以追溯到量子力学，在量子力学中，位置和动量等经典可观测物不再交换。近年来，人们认识到这样的非交换空间保留了丰富的拓扑和几何，这些拓扑和几何首先表达在k理论和k同调中，以及该理论的更精细的方面。随着量子群及其量子齐次空间的出现，该主题也从更代数的方面进行了探讨。这门学科的现代形式也与纯数学和数学物理的几个不同领域的发展联系在一起。在数学中，这些包括与分析、数论、范畴论和表示论的富有成效的互动。在数学物理方面，发展包括量子霍尔效应、粒子物理标准模型和量子场论重整化的应用、具有非交换坐标的时空模型。非交换几何也自然地出现在弦/ m理论中。该计划将致力于汇集这些不同的流和实例的非交换几何，以及确定新的新兴方向。该计划的三个主要主题将在2006年7月、9月和12月的研讨会上反映出来，分别包括非交换几何和循环上同调、非交换几何和基础物理以及非交换几何的新方向。

This award is for partial support of US participants in a semester long workshop at the Newton Institute, Cambridge, UK, in noncommutative geometry for the period July 24 through December 22, 2006. Noncommutative geometry aims to carry over geometrical concepts to a new class of spaces whose algebras of functions are no longer commutative. The central idea goes back to quantum mechanics, where classical observables such as position and momenta no longer commute. In recent years it has become appreciated that such noncommutative spaces retain a rich topology and geometry expressed first of all in K-theory and K-homology, as well as in finer aspects of the theory. The subject has also been approached from a more algebraic side with the advent of quantum groups and their quantum homogeneous spaces. The subject in its modern form has also been connected with developments in several different fields of both pure mathematics and mathematical physics. In mathematics these include fruitful interactions with analysis, number theory, category theory and representation theory. In mathematical physics, developments include the quantum Hall effect, applications to the standard model in particle physics and to renormalization in quantum field theory, models of spacetimes with noncommuting coordinates. Noncommutative geometry also appears naturally in string/M-theory. The program will be devoted to bringing together these different streams and instances of noncommutative geometry, as well as identifying new emerging directions. Three main themes of the program will be reflected in workshops in July, September and December of 2006, covering noncommutative geometry and cyclic cohomology, noncommutative geometry and fundamental physics, and new directions in noncommutative geometry respectively.