Trends in noncommutative geometry

摘要奖：DMS-0728322主要研究人员：Boris Tsygan，Dmitry Tamarkin该奖项为西北大学在强调非对易几何方面的会议提供了部分支持。它的目的是探索非对易几何中几个方向之间的相互作用：指数定理、非对易微积分和算符、与数学物理、数论和几何语言程序的关系。一个重要的目标是突出芝加哥地区强有力的主题(代数几何、动机、微局部分析、表示理论、几何语言程序、镜像对称)，并调查它们与传统上被认为是非对易几何的工作的联系，反之亦然。例如，将几何朗兰兹程序和实群表示理论中使用的LOOP空间的一种代数方法与与代数拓扑学和数学物理有关的拓扑弦理论以及p-进群的表示理论的一种方法进行了比较。在这些学科之间建立了一种关系，特别是严格拓扑学，和当前关于动机的研究。给出了非对易几何与镜像对称之间猜想关系的几种方法。非对易几何是将经典几何推广到坐标x，y等不再交换的“空间”，即恒等式xy=yx不一定是真的。这种“空间”在数学和物理学中比比皆是。例如，非对易是量子力学中海森伯格测不准原理的数学表现：如果x代表粒子的位置，y代表其动量，你不可能同时知道x和y的精确值。在更数学的背景下，非对易是这样一种事实的表达，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。发生非对易的数学情形早已为人所知，但将它们视为“非对易空间”并用几何方法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。会议的目的是比较非对易几何的几种方法，以及它与其他数学和物理领域的联系。您可以在会议网站athttp://www.math.northwestern.edu/~tsygan/conf上找到更多信息

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