Noncommutative geometry, microlocal analysis, index theorems and symplectic geometry

AbstractAward：DMS-0906391首席研究员：Boris Tsygan该项目致力于研究指数理论和辛几何中的微局部方法。该项目的一部分涉及非交换微分学，这是上述中心工具之一。通过非交换微分学，我们的意思是研究的代数结构所产生的标准微分流形上，定义在这样一种方式，是有效的代数的功能在一个流形上，并在这样一种方式，是有效的，如果这个代数是取代任何代数，交换或不。这导致了研究代数的Hochschild链和cochaincomplex的代数结构;这些复杂的发挥作用的微分形式和多向量领域，一个发展的工作Dolgushev-Tamarkin-Tsygan，Kontsevich-Soibelman，Costello，Lurie和其他作者。将应用非可交换微积分技术获得新的指标定理和有关椭圆系统上同调行列式的定理，推广Beilinson最近的结果。此外，本计画将研究辛流形上的一个新对象，我们称之为振荡模。这些对象是流形上函数的模过变形代数，但具有使其范畴更接近于超代数范畴的额外结构。这种额外的结构在很大程度上是由偏微分方程中的微局部方法，特别是WKB方法激发的。这是一系列试图理解镜像对称微观局部的工作的一部分（Bressler-Soibelman，Kapustin-Witten，Nadler-Zaslow，Gukov-Witten，Tamarkin）。在这个项目中有三个相关的部分：非对易微积分，指数理论和振荡模。非对易微积分是一种理论，它描述了标准微分学的一部分，这种理论可以应用于比普通空间更一般的情况，特别是当我们想象的“空间”上的坐标不再需要方程xy=yx时。指数理论是微分方程理论的一部分，它用底层空间的拓扑来表示方程的解的个数。振荡模是几何空间上的对象，描述量子粒子在这些空间上的运动（换句话说，它们将所讨论的空间视为量子力学系统的相空间）。这些物体被用来研究这些空间的其他不变量，也是物理上的动机，而且要复杂得多;它们被称为A-膜，描述了量子弦在我们空间中的运动。 它们被称为A-膜，是数学物理学和几何学中最新研究的主题。最后，让我们解释一下为什么非对易微积分在微分方程和量子力学中的应用是很自然的。事实上，如果x和y是函数的基本运算，例如，乘以某个函数和微分，那么xy和yx就不同，在这个意义上，如果你以不同的顺序应用它们，你会得到不同的结果。类似地，在量子力学中，如果x是粒子的位置，y是它的动量，那么xy不同于yx;这是海森堡不确定性原理的数学表达。

AbstractAward: DMS-0906391Principal Investigator: Boris TsyganThis project is devoted to studying microlocal methods in indextheory and in symplectic geometry. A part of the project dealswith noncommutative differential calculus which is one of thecentral tools in the above. By noncommutative differentialcalculus we mean the study of algebraic structures arising fromthe standard differential calculus on manifolds, defined in termsin such a way that is valid of the algebra of functions on amanifold and in such a way that is valid if this algebra isreplaced by any algebra, commutative or not. This leads to thestudy of algebraic structures on Hochschild chain and cochaincomplexes of an algebra; those complexes play the role ofdifferential forms and multivector fields, a development of worksof Dolgushev-Tamarkin-Tsygan, Kontsevich-Soibelman, Costello,Lurie and other authors. The techniques of noncommutativecalculus will be applied to obtain new index theorems andtheorems about the determinant of the cohomology of ellipticsystems, generalizing recent results of Beilinson. Also, theproject will study a new object on a symplectic manifold that wecall an oscillatory module. These objects are modules overdeformed algebras of functions on a manifold, but with extrastructures that make their category much closer to the Fukayacategory. This extra structure is largely motivated by microlocalmethods in partial differential equations, in particular the WKBmethod. This is a part of a series of works that try tounderstand mirror symmetry microlocally (Bressler-Soibelman,Kapustin-Witten, Nadler-Zaslow, Gukov-Witten, Tamarkin).There are three related parts in the project: noncommutativecalculus, index theory, and oscillatory modules. Noncommutativecalculus is a theory that describes parts of the standarddifferential calculus in such a way that it can be applied insituations more general than that of an ordinary space, inparticular when, for coordinates on our imaginary "space", theequation xy=yx is no longer required. Index theory is a part ofthe theory of differential equations that expresses the number ofsolutions of an equation in terms of the topology of theunderlying space. Oscillatory modules are objects on geometricspaces that describe the motion of quantum particles on thesespaces (in other words, they treat the space in question as thephase space of a quantum mechanical system). These objects areintended to be applied to study other invariants of these spaces,also physically motivated and much more complicated; they arecalled A-branes and describe the motion of a quantum string onour space. They are called A-branes and are subject of muchcurrent research in mathematical physics and geometry. Let usfinish by explaining why noncommutative calculus is natural forapplications both in differential equations and in quantummechanics. Indeed,if x and y are basic operations withfunctions, for example multiplication by some function anddifferentiation, then xy differs from yx, in the sense that, ifyou apply them in different orders, you get differentresults. Similarly, in quantum mechanics, if x is the position ofa particle and y is its momentum, then xy differs from yx; thisis a mathematical expression of the Heisenberg uncertaintyprinciple.