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。

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.