Trends in noncommutative geometry

奖项：dms -0728322首席研究员：Boris Tsygan， Dmitry tamarkin该奖项为西北大学在重点年度结束时举行的非交换几何会议提供了部分支持。它的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、非交换微积分和操作数、与数学物理、数论和几何朗朗兹程序的关系。一个重要的目标是突出在芝加哥地区很强的学科（代数几何、动机、微局部分析、表征理论、几何朗兰兹程序、镜像对称），并研究它们与传统上被认为是非对易几何的作品之间的联系，反之亦然。例如，将用于几何朗兰兹规划和实数群表示理论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进群表示理论的方法进行了比较。这些学科，特别是弦拓扑，与当前的动机研究之间建立了联系。提出了几种方法来推测非交换几何与镜像对称之间的关系。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。等式xy=yx不一定成立。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：如果表示粒子的位置，y表示它的动量，你不可能知道x和y的精确值。在更数学的背景下，非交换性是一个事实的表达，如果你对一个物体应用两个变换，结果取决于你做它的顺序。发生非交换性的数学情形早已为人所知，但将其视为“非交换空间”并用几何方法研究它们的想法是相对较新的。这种方法的一个成功之处是通过允许空间非对易，彻底简化了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，以及它与其他数学和物理领域的联系。您可以在会议网站https://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