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

非交换几何（NCG）是一个应用代数（包括非交换代数）解决几何问题的数学领域。非交换代数中的表达式对元素的阶很敏感。例如，如果你洗一副牌，然后剪切它，如果你剪切它，然后洗它，结果是不一样的。流形局部看起来像点，线，平面等。例如，气球和轮胎管是2维流形，因为你可以通过缝合一些平面状的碎片来构造它们。另一方面，一个非交换流形通常有两个部分，一个是流形，另一个是流形上的一个关系，它可以识别它的一些点。显然，任何流形都是非交换流形，但反之则不然。流形上的叶理把它切割成维数较低的流形。例如，一个洋葱是一个三维流形，它被分解成它的叶子，叶子都是球形的，因此是二维的。同样，卷心菜也分解成叶子。从数学上讲，洋葱和卷心菜是一样的，但它们的叶子不相等。通过识别每个叶的所有点，我们得到一个非交换流形。叶子在数学和物理学的许多领域都有应用。叶理的研究历史悠久，自20世纪60年代以来，其表征一直是开放的。在这一领域应用了不同的理论，并取得了部分成功。然而，仍然存在许多根本性的开放问题。我们计划通过非交换几何的方法来解决这些开放的问题。我们的主要工具是霍普夫代数和霍普夫循环上同调。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。

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.