Noncommutative Geometry at the Newton Institute

该奖项是为了表彰在英国剑桥牛顿研究所举办的为期一学期的非对易几何研讨会的美国参与者的部分支持，该研讨会的时间为2006年7月24日至12月22日。非对易几何的目的是将几何概念推广到函数代数不再是对易的一类新空间。核心思想回到了量子力学，在量子力学中，位置和动量等经典可观测变量不再相互作用。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，首先是在K-理论和K-同调中，以及在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，这个问题也从更代数的方面得到了解决。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相联系。在数学方面，这些包括与分析、数论、范畴理论和表征理论的卓有成效的互动。在数学物理方面，发展包括量子霍尔效应，应用于粒子物理中的标准模型和量子场论中的重整化，具有非对易坐标的时空模型。非对易几何也自然地出现在弦/M-理论中。该计划将致力于将非对易几何的这些不同的流和实例结合在一起，并确定新的新兴方向。该计划的三个主要主题将反映在2006年7月、9月和12月的研讨会上，分别涉及非对易几何和循环上同调、非对易几何和基础物理以及非对易几何的新方向。

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.