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

非交换几何（NCG）是应用代数（包括非交换代数）解决几何问题的一个数学领域。非交换代数中的表达式对其元素的顺序很敏感。例如，如果你洗牌，然后切牌，结果是不一样的，如果你切牌，然后洗牌。流形在局部看起来像点、线、平面等。例如，气球和轮胎管是2维的流形，因为你可以通过将一些平面状的碎片拼接在一起来构造它们。另一方面，一个非交换流形通常由两个部分组成，一个流形和一个关系，这个关系确定了流形上的一些点。很明显，任何流形都是不可交换的，反之则不然。流形上的叶化将流形切成低维的流形。例如，一个三维流形的洋葱被分解成它的叶子，这些叶子都是球形的，因此是二维的。同样，卷心菜也被分解成叶子。从数学上讲，洋葱和卷心菜是一样的，但它们的叶子不相等。通过识别每个叶上的所有点，我们得到了一个非交换流形。叶理在数学和物理的许多领域都有应用。对叶理的研究非常古老，它们的特征自20世纪60年代以来一直开放。在这一领域应用了不同的理论，并取得了部分成功。然而，仍有许多根本性的问题有待解决。我们计划用非交换几何的方法来解决这些开放问题。我们的主要工具是Hopf代数和Hopf循环上同。Hopf代数是对称的非交换对应物。Hopf循环上同调定义了对称非交换流形的不变量。这项研究不仅有利于叶形理论的发展，而且使我们能够将叶形及其特征类应用于数论、量子场论、结理论和变形理论等许多其他数学领域。例如，圆桌的对称是平面上所有围绕原点的旋转。然而，矩形表的对称性是平面上坐标轴的反射。例如可实现性猜想，它推测Gelfand-Fuks上同调中的每一类都存在一个不同的叶状，这一猜想还远未得到解决。经典几何中的对称是由群控制的，而研究非交换空间需要Hopf代数（量子群）作为群的推广。Hopf循环上同调是在Hopf代数对称下提供代数不变量的理论。这个理论是由cones - moscovic在1998年发明的，从那以后，他和他的合作者一直在发展这个理论。在这个项目的第一阶段，我们已经应用Hopf代数和它们的上同调来发展新的理论来表征叶理。其中一些理论在叶化理论中没有对应的理论。例如，通过稳定-反叶特-德林菲尔德模的扭转仅仅是由于Hopf循环上同。我们最近对这一问题的研究使Hopf循环理论成为叶形特征类的一个新家。我们的目标是在Hopf循环理论的帮助下继续重新表述和解决叶化问题。从非交换几何的观点出发，将Hopf循环上同应用于叶的特征类的研究。这是我们对这个主题的另一项长期研究，我们计划通过它来重新表述叶分及其不变量。我们的最终目标是证明可实现性和可识别性的猜想。到目前为止，我们的研究已经证明了Hopf代数是非交换空间中对称和无穷小的合适来源。一个可以应用非交换几何的所有方面的非交换空间是叶状叶的空间，微分几何对它无能为力。在另一个并行但不远的项目中，我们计划从量子群上的Yetter-Drinfled模的范畴中推导出结点的量子不变量。我们想把最近关于包络代数的结果推广到量子代数。非交换几何是由算子代数和微分几何的结合发展起来的。这与微分几何相反，在微分几何中，空间是点的集合，而它们上的函数是次要对象。Gelfand-Naimark定理建立了交换C*-代数与局部紧化Hausdorff空间的等价性，可以很好地理解这种角色交换。然而，作为一个重要的公理，我们不剥夺非交换代数作为“空间”的坐标代数。在这个项目中，我们应用并发展了Hopf环上同调的各种概念和对象，包括：横向几何的Hopf代数、稳定的反叶特模、杯积、扭曲的环共环和局部指数公式。例如，考虑洗牌和切割一副牌。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。通过经典几何，你无法看到图像。NCG为您提供了像素位置的笔记本。在这个例子中，分散的拼图和笔记本就像一个非交换空间的例子；而解决的谜题则代表了一个经典的空间。为了理解对称的意义，你可以在一张纸上画一个正方形。然后，薄片的移动对正方形没有影响，由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给出。正方形的对称组就是这两种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.