Trends in noncommutative geometry

非交换几何的趋势

• 批准号: 0728322
• 负责人: Boris Tsygan
• 金额: --
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0728322&HistoricalAwards=false
• 关键词: Trends; noncommutative; geometry

AbstractAward: DMS-0728322Principal Investigator: Boris Tsygan, Dmitry TamarkinThis award provides partial support for a conference at the endof the emphasis year on noncommutative geometry at NorthwesternUniversity. Its aim was to explore the interaction betweenseveral directions in noncommutative geometry: index theorems,noncommutative calculus and operads, relations to mathematicalphysics, to number theory, and to the geometric Langlandsprogram. An important objective was to highlight the subjectsthat are strong in the Chicago area (algebraic geometry, motives,microlocal analysis, representation theory, geometric Langlandsprogram, mirror symmetry) and investigate their links to theworks in what is more traditionally considered noncommutativegeometry, and vice versa. For example, an algebraic approach toloop spaces that is used in the geometric Langlands program andin representation theory of real groups was compared to thetopological string theory which is related to algebraic topologyand mathematical physics, as well as to an approach to therepresentation theory of p-adic groups. A relation wasestablished between these subjects, in particular stringtopology, and current research on motives. Several approaches toa conjectural relationship between noncommutative geometry andmirror symmetry were presented.Noncommutative geometry is an extension of the classical geometryto "spaces" in which coordinates x, y, etc. no longer commute,i.e. where the identity xy=yx is not necessarily true. Such"spaces" abound in mathematics and physics. For example,noncommutativity is the mathematical manifestation of theHeisenberg uncertainty principle in quantum mechanics: if xstands for a position of a particle and y for its momentum, youcannot know precise values of both x and y. In a moremathematical context, noncommutativity is an expression of thefact that, if you apply two transformations to an object, theresult depends on the order in which you do it. Mathematicalsituations where noncommutativity occurs have been known for along time but the idea to treat them as "noncommutative spaces"and to study them by geometric methods is relatively recent. Oneof the successes of this approach is a radical simplification ofthe standard model in quantum field theory by means of allowingthe space to be noncommutative. The aim of the conference was tocompare several approaches to noncommutative geometry, as well asits links to other areas of mathematics and physics. You can findmore information on the conference website athttp://www.math.northwestern.edu/~tsygan/conf

AbstractAward：DMS-0728322首席研究员：Boris Tsygan，Dmitry Tamarkin该奖项为西北大学非对易几何重点年度结束时的会议提供了部分支持。其目的是探索非交换几何中几个方向之间的相互作用：指标定理、非交换微积分和运算、与数学物理、数论和几何朗兰兹纲领的关系。一个重要的目标是突出的subjectsthat是强大的芝加哥地区（代数几何，动机，微观局部分析，表示理论，几何Langlandsprogram，镜像对称），并调查他们的联系，以工程在什么是更传统地认为noncommutativegeometry，反之亦然。例如，一个代数的方法，以环空间中使用的几何朗兰兹计划和表示理论的真实的集团进行了比较，拓扑弦理论，这是有关代数拓扑和数学物理，以及一种方法，representation理论的p-adic集团。这些主题之间的关系，特别是弦拓扑学，和目前的研究动机。非对易几何是经典几何在坐标x，y等不再对易的“空间”中的推广，即恒等式xy=yx不一定为真。这样的“空间”在数学和物理学中比比皆是。例如，非对易性是量子力学中海森堡测不准原理的数学表现：如果x代表粒子的位置，y代表它的动量，你不可能知道x和y的精确值。在一个更加数学化的语境中，非对易性是这样一个事实的表达，即如果你对一个物体进行两次变换，结果取决于你进行变换的顺序。非对易性发生的数学情形已经被人们所知沿着，但是把它们当作“非对易空间“并用几何方法来研究它们的想法是相对较新的。这种方法的成功之一是通过允许空间是非对易的，从根本上简化了量子场论中的标准模型。会议的目的是比较几种方法，以非交换几何，以及它的联系，其他领域的数学和物理。您可以在会议网站http：//www.example.com上找到更多信息www.math.northwestern.edu/~tsygan/conf