Noncommutative geometry, microlocal analysis, index theorems and symplectic geometry

摘要奖：DMS-0906391首席研究员：Boris Tsygan这个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到非对易微积分，这是上述课程的核心工具之一。我们所说的非交换微积分是指研究流形上的标准微积分所产生的代数结构，其定义方式是流形上的函数代数是有效的，并且如果这个代数被任何代数取代，无论是交换的还是非交换的，这种方式都是有效的。这导致了对代数的Hochschild链和Cochain复形上的代数结构的研究；这些复形起到了微分形式和多向量场的作用，发展了Dolgushev-Tamarkin-Tsygan，Kontsevich-Soibelman，Costello，Lurie和其他作者的工作。利用非对易微积分的技巧，推广了Beilinson的最新结果，得到了椭圆系的上同调行列式的新指标定理和行列式定理。此外，该项目还将研究辛流形上的一个新对象，该对象称为振荡模。这些对象是流形上函数的模超变形代数，但具有使它们的范畴更接近Fukayac范畴的外部结构。这种额外的结构在很大程度上是由偏微分方程组的微域方法，特别是WKB方法所推动的。本文是Bressler-Soibelman，Kapustin-Witten，Nadler-Zaslow，Gukov-Witten，Tamarkin等一系列微观局部研究工作的一部分。该项目包括三个相关部分：非对易微积分、指数理论和振荡模。非对易微积分是一种描述部分标准微积分的理论，它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组理论的一部分，它用基础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物体，描述了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)。这些物体打算用来研究这些空间的其他不变量，也是由物理驱动的，而且要复杂得多；它们被称为A-膜，描述了量子弦在我们的空间中的运动。它们被称为A膜，是目前数学物理和几何研究的对象。最后，让我们解释一下为什么非对易微积分在微分方程式和量子力学中的应用是自然的。事实上，如果x和y是有函数的基本运算，例如乘以某个函数和微分，那么xy与yx的不同之处在于，如果你以不同的顺序应用它们，你会得到不同的结果。同样，在量子力学中，如果x是粒子的位置，y是粒子的动量，那么xy不同于yx；这是海森堡测不准原理的数学表达式。

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.