Homological algebra of quantum invariants in dimension four

米哈伊尔·G·霍瓦诺夫该项目旨在构建4维物体的量子不变量。它是基于作者最近发现的关于3-球面中链环的双分次上同调理论。上同调群的欧拉特征等于琼斯多项式。我们想将这一理论扩展到将密码学联系起来。上同调群的不变量是分配给上同调群边界的上同调群之间的同态。此外，这一理论应该扩展到纠缠和纠缠共治。对于缠绕，我们将在与缠绕的边界相关联的三角化类别之间关联函子，并将函子之间的自然转换关联到缠绕共边。这些三角范畴将与单李代数上最高权的模范畴有关，也与某些Frobenius代数上的模范畴有关，例如分圆Hecke代数。此外，我们还将寻找环和三维流形的其他量子不变量的上同调理论Lift，包括Alexander和HOMFLY多项式以及Witten-Reshetikhin-Turaev不变量。N维流形是局部看起来像n维空间的对象。圆可以用点附近的切线来近似，它是一维流形(简称n流形)。不过，全球结构将圆圈与直线区分开来。曲面提供了两个流形的例子。结果表明，一维流形和二维流形很容易分类，而高维流形则很难分类。这是一个定理，在大于3的维度中，不可能有令人满意的分类，而拓扑学家似乎相当接近为3-流形找到一个分类。给定一对流形，很难确定它们是否同构。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。大多数时候，数字是不同的，告诉我们流形也是不同的。第三维的特别之处在于有大量这样的不变量。此外，这些不变量将三维流形拓扑与深代数结构联系起来。有迹象表明，不变量可以提升到下一个维度，四维流形的不变量，我的目标是找到它们，并将它们与由某些偏微分方程解产生的四维流形的解析不变量进行比较。

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.