Hopf Cyclic Cohomology, Characteristic Classes of Foliations, and Quantum Invariant of Knots.

非对易几何(NCG)是应用代数(包括非对易代数)来解决几何问题的数学领域。非交换代数中的表达式对其元素的顺序很敏感。例如，如果你洗一副牌，然后把它切开，如果你把它切开，然后洗牌，结果就不一样了。歧管在局部看起来像点、线、面等。例如，气球和轮胎内胎是2维的歧管，因为你可以通过将几个平面状的片段缝合在一起来构建它们。另一方面，非交换流形通常由两部分组成，即流形和流形上标识某些点的关系。显然，任何流形都是非对易流形，但反之亦然。流形上的叶状结构将流形分割成较低维度的流形。例如，一个三维流形的洋葱被分解成它的叶子，这些叶子都是球状的，因此是二维的。同样，卷心菜也会分解成叶子。从数学上讲，洋葱和卷心菜是一样的，但它们的叶不相等。通过识别每个叶的所有点，我们得到了一个非对易流形。复数在数学和物理的许多领域都有应用。对叶理的研究由来已久，自20世纪60年代以来，叶理的特征一直是开放的。已经有不同的理论应用于这一领域，并取得了部分成功。然而，仍然存在许多根本性的开放问题。我们计划通过非对易几何的方法来解决这些公开问题。我们的主要工具是Hopf代数和Hopf循环上同调。Hopf代数是对称的非交换对应。Hopf循环上同调定义了对称非交换流形的不变量。这一研究不仅对叶层理论有所裨益，而且使我们能够将叶层及其特征类应用于数论、量子场论、纽结理论和形变理论等许多数学领域。例如，圆桌的对称性就是绕平面原点的所有旋转。然而，矩形工作台的对称性是反射到平面中的坐标轴。例如，可实现性猜想推测，对于Gelfand-Fuks上同调中的每一类，都存在一个明显的叶状结构，这一猜想仍然远未解决。经典几何中的对称性是由群决定的，而研究非对易空间需要Hopf代数(量子群)作为群的推广。Hopf循环上同调是一种在Hopf代数的对称性下给出代数不变量的理论。该理论由康奈斯-莫斯科维奇于1998年发明，自那以来一直由提出者及其合作者发展。在这个项目的第一阶段，我们应用Hopf代数和它们的上同调来发展新的理论来刻画叶状结构。其中一些理论在叶状结构理论中是没有对应的。例如，通过稳定的反Yetter-Drinfeld模进行的扭曲仅仅是由于Hopf循环上同调。我们最近对这一课题的研究使Hopf旋回理论成为叶理特征类的新归宿。我们的目标是继续借助霍普夫循环理论重新表述和解决叶理问题。从非对易几何的角度出发，我们提出将Hopf循环上同调应用于叶状结构的特征类的研究。这是我们对这一主题的另一项长期研究，我们计划通过它来重新表述叶及其不变量。我们的最终目标是证明可实现性和可识别性猜想。到目前为止，我们的研究已经证实，Hopf代数是非对易空间中对称和无穷小的合适来源。可以运用非对易几何的所有方面的非对易空间之一是叶层的叶空间，在该空间上微分几何无能为力。在另一个平行但并不遥远的项目中，我们计划从量子群上的Yetter-Drfled模范畴推导出纽结的量子不变量。我们想把我们关于包络代数的最新结果推广到量子代数。非对易几何是由算子代数和微分几何的结合发展而来的。这与微分几何不同，在微分几何中，空间是点的集合，而点上的函数是次要对象。这种角色的交换被Gelfand-Naimark定理很好地理解，该定理建立了交换C*-代数和局部紧Hausdorff空间的等价性。然而，作为一个重要的公理，我们并不剥夺非对易代数是一个“空间”的坐标代数。在这个项目中，我们应用并发展了Hopf循环上同调的各种概念和对象，包括：横向几何的Hopf代数、稳定的反Yetter-Drfled模、杯积、扭曲循环上循环和局部指数公式。例如，考虑一副牌的洗牌和切割。然后是切割，然后洗牌和洗牌然后切割之间的区别。假设有一个复杂的谜题。通过古典几何学，你看不到这幅画。NCG为您提供像素位置的笔记本。在这个例子中，散乱的拼图和笔记本类似于一个非对易空间的例子；而解开的拼图则代表一个经典空间。要了解对称的意义，你可以在一张纸上画一个正方形。然后，对正方形没有影响的薄片的移动是通过围绕正方形中心旋转90度和相对于将正方形分成两个相等矩形的线的反射来给出的。正方形的对称群包括这两个简单动作的重复和组合。

Noncommutative Geometry (NCG) is an area of mathematics which applies algebras, including non-commutative ones, in solving geometric problems. Expressions in non-commutative algebras are sensitive to the order of their elements. For example, if you shuffle a deck of cards and then cut it, the result is not the same if you cut it and then shuffle it. Manifolds look locally like points, lines, planes, etc. For example, balloons and tire tubes are manifolds of dimension 2 because you may construct them by stitching together a few plane-like pieces. On the other hand, a non commutative manifold usually comes with two parts which are a manifold and a relation on it that identifies some of its points. It is obvious that any manifold is a non-commutative one but not vice versa. A foliation on a manifold slices it into manifolds with a lower dimension. For example, an onion which is a three dimensional manifold is decomposed into its leaves which are all sphere-like and hence two dimensional. Similarly a cabbage is also decomposed into its leaves. Mathematically, onion and cabbage are the same but their foliations are not equal. By identifying all points of each leaf we obtain a non-commutative manifold. Foliations have applications in many areas of mathematics and physics. Study of foliations is very old and their characterization has been open since 1960s. There have been different theories applied in this area with partial successes. However there are still many fundamental open problems. We plan to solve these open problems via methods of non-commutative geometry. Our primary tools are Hopf algebras and Hopf cyclic cohomology. Hopf algebras are the non-commutative counterpart of symmetries. Hopf cyclic cohomology defines invariants for symmetric non-commutative manifolds. This study not only will benefit the theory of foliations but also enables us to apply foliations and their characteristic classes in many other areas of mathematics such as number theory, quantum field theory, knot theory, and deformations theory. For example, symmetry of a round table is all rotations around the origin in the plane. However the symmetry of a rectangle table is reflections to the coordinate axes in the plane. For example the realizablity conjecture, which speculates that for each class in the Gelfand-Fuks cohomology there exists a distinguished foliation, is still far from being solved. Symmetry in classical geometry is governed by groups, while to study noncommutative spaces one needs Hopf algebras(quantum groups) as a generalization of groups. Hopf cyclic cohomology is a theory which provides invariants of algebras under symmetry of Hopf algebras. The theory was invented by Connes-Moscovic in 1998 and since then has been developed by the proposer and his collaborators. In the first phase of this project we have applied Hopf algebras and their cohomology to develop new theories for characterization of foliations. Some of these theories have no counterpart in the theory of foliations. For instance the twisting via Stable-Anti-Yetter-Drinfeld modules is merely due to Hopf cyclic cohomology. Our recent investigations on this subject make Hopf cyclic theory a new home for characteristic classes of foliations. Our goal is to continue reformulating and solving foliations problems with the help of Hopf cyclic theory. We propose to apply Hopf cyclic cohomology in the study of characteristic classes of foliations from the Noncommutative Geometry point of view. This is another piece of our long term research on the subject by which we plan to reformulate foliations and their invariants. Our ultimate goal is to prove realizablility and recognizability conjectures. Our research so far has established the fact that Hopf algebras are suitable sources of symmetry and infinitesimals for noncommutative spaces. One of the noncommutative spaces on which one can exercise all aspects of Noncommutative Geometry is the space of leaves of a foliation, on which Differential Geometry is powerless. In another parallel but not far project we plan to derive quantum invariant of knots from the category of Yetter-Drinfled modules over quantum groups. We would like to extend our recent results on the enveloping algebras to quantum algebras. Noncommutative geometry developed out of coalescence of operator algebras and differential geometry. This is in contrast to Differential Geometry, where spaces are sets of points and functions on them are secondary objects. This exchange of roles is well understood by the Gelfand-Naimark theorem which establishes the equivalence of commutative C*-algebras and locally compact Hausdorff spaces. However, as a crucial axiom we do not deprive noncommutative algebras of being coordinates algebras of a "spaces". In this project we apply and also develop a wide variety of concepts and objects of Hopf cyclic cohomology including: Hopf algebras of transverse geometries, stable-anti-Yetter-Drinfled modules, cup products, twisted cyclic cocycles, and local index formula.For example consider shuffling and cutting of a deck of cards. Then there is a difference between cutting then shuffling and shuffling and then cutting. Assume that there is a complicated puzzle. Via classical geometry you cannot see the picture. NCG provides you with the notebook of the locations of the pixels. In this example the scattered puzzle together with the notebook resembles an example of noncommutative spaces; while the solved puzzle represents a classical space. To see the meaning of symmetry you may draw a square in a sheet. Then the moves of the sheet with no effect on the square are given by the rotation around the center of the square by 90 degree and the reflection with respect to a line that divides the square into two equal rectangles. The group of symmetries of square comprises the repetition and combination of theses two simple moves.