Homological algebra of quantum invariants in dimension four

四维量子不变量的同调代数

• 批准号: 0104139
• 负责人: Mikhail Khovanov
• 金额: $ 5.71万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104139&HistoricalAwards=false
• 关键词: Homological; algebra; quantum; invariants; dimension

DMS-0104139Mikhail G. KhovanovThe project aims to construct quantum invariants of 4-dimensional objects. It is based on the author's recent discovery of a doubly-graded cohomology theory of links in the 3-sphere. The Euler characteristic of the cohomology groups is equal to the Jones polynomial. We would like to extend this theory to link cobordisms. The invariant of a cobordism will be a homomorphism between cohomology groups assigned to the boundaries of the cobordism. Furthermore, the theory should extend to tangles and tangle cobordisms. To a tangle we'll associate a functor between triangulated categories associated to the boundaries of the tangle, and to a tangle cobordism a natural transformation between functors. These triangulated categories will be related to highest weightcategories of modules over simple Lie algebras, as well as categories of modules over certain Frobenius algebras, such as cyclotomic Hecke algebras. In addition, we will look for cohomology theories lifts of other quantum invariants of links and 3-manifolds, including the Alexander and HOMFLY polynomials and Witten-Reshetikhin-Turaev invariants. An n-dimensional manifold is an object that locally looks like an n-dimensional space. A circle can be approximated by a tangent line in the neighbourhood of a point, and is a one-dimensional manifold (n-manifold, for short). The global structure distinguishes the circle from the line, though. Surfaces provide examples of two-manifolds. It turns out that one and two-manifolds are easy to classify, while in higher dimensions classification is hard. It is a theorem that in dimensions greater than three there can be no satisfactory classification, and topologists seem to be fairly close to finding one for three-manifolds. Given a pair of manifolds, it is a tough question to decide whether or not they are isomorphic. One approach is to extract some tangible invariant out of a manifold, such as a number, or a polynomial, and then compare these numbers. Most of the times the numbers are different and tell us that the manifolds are different, too. Dimension three is special in that there is a wealth of such invariants. These invariants, moreover, link three-manifold topology with deep algebraic structures. There are indications that the invariants can be lifted to the next dimension, to invariants of four-manifolds, and my goal is to find them and compare to analytical invariants of 4-manifolds that arise from solutions of certain partial differential equations.

Mikhail G. Khovanov该项目旨在构建四维物体的量子不变量。 它是基于作者最近发现的一个双阶上同调理论的链接在3-球。上同调群的欧拉特征标等于琼斯多项式。 我们想把这个理论扩展到链接配边。配边的不变量是指配边边界上的上同调群之间的同态。此外，该理论应该扩展到缠结和缠结配边。对于一个tangle，我们将把一个与tangle边界相关联的三角化范畴之间的函子关联起来，对于一个tangle cobordism，我们将把一个函子之间的自然变换关联起来。这些三角范畴将与单李代数上的最高权模范畴有关，也与某些Frobenius代数上的模范畴有关，例如分圆Hecke代数。此外，我们将寻找其他量子不变量的链接和3-流形，包括亚历山大和HOMFLY多项式和Witten-Reshetikhin-Turaev不变量的上同调理论升降机。 n维流形是一个局部看起来像n维空间的对象。一个圆可以用一个点附近的切线来近似，并且是一个一维流形（简称n流形）。然而，全局结构将圆与线区分开来。曲面提供了双流形的例子。 事实证明，一维和二维流形很容易分类，而在高维分类是困难的。这是一个定理，在尺寸大于三，不可能有令人满意的分类，拓扑学家似乎相当接近找到一个三流形。给定一对流形，判断它们是否同构是一个很坚韧的问题。一种方法是从流形中提取一些有形的不变量，例如数字或多项式，然后比较这些数字。大多数情况下，数字是不同的，这告诉我们流形也是不同的。三维空间的特殊之处在于有大量这样的不变量。此外，这些不变量将三流形拓扑与深层代数结构联系起来。有迹象表明，不变量可以提升到下一个维度，四流形的不变量，我的目标是找到它们，并与某些偏微分方程的解所产生的四流形的分析不变量进行比较。