Trends in noncommutative geometry

摘要奖项：DMS-0728322 首席研究员：Boris Tsygan、Dmitry Tamarkin 该奖项为西北大学非交换几何重点年末的会议提供部分支持。其目的是探索非交换几何中几个方向之间的相互作用：指数定理、非交换微积分和运算、与数学物理、数论以及几何朗兰兹纲领的关系。一个重要的目标是突出芝加哥地区的优势学科（代数几何、动机、微局域分析、表示论、几何朗兰兹纲领、镜像对称），并研究它们与传统上被认为是非交换几何的著作之间的联系，反之亦然。例如，将几何朗兰兹纲领和实群表示论中使用的环空间代数方法与与代数拓扑和数学物理相关的拓扑弦理论以及 p 进群表示论方法进行了比较。这些学科（特别是弦拓扑学）与当前动机研究之间建立了联系。提出了几种非交换几何与镜像对称之间猜想关系的方法。非交换几何是经典几何对坐标 x、y 等不再交换的“空间”的扩展，即其中恒等式 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，非交换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 代表粒子的动量，那么你就无法知道 x 和 y 的精确值。在更数学化的背景下，非交换性表达了这样一个事实：如果你对一个对象应用两次变换，结果取决于你进行变换的顺序。出现非交换性的数学情况早已为人所知，但将它们视为“非交换空间”并通过几何方法研究它们的想法相对较新。这种方法的成功之一是通过允许空间不可交换来彻底简化量子场论中的标准模型。会议的目的是比较非交换几何的几种方法，以及它与数学和物理其他领域的联系。您可以在会议网站上找到更多信息：http://www.math.northwestern.edu/~tsygan/conf

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