Homological algebra and topology in three and four dimensions

该项目旨在开发和理解三维和四维拓扑对象的同调不变量。这些不变量包括Donaldson-Floer和Seiberg-Witten理论，以及几个关于链环的双阶同调理论。这些理论具有亚历山大和琼斯多项式作为它们的欧拉特征，以及量子sl（3）链不变量。我们想对每个复单李代数g构造一个链环的双阶同调理论，链环的分支由g的不可约表示着色。该理论的欧拉特征应该是与g的量子形变相关的有色链接的量子不变量，并且该理论应该是函子的（扩展到链接的协边）。我们的其他目标包括更好地理解现有理论之间的关系，并调查的类别时出现的链接同源性理论扩展到缠结。三维和四维中的拓扑对象具有特殊的性质以及与代数和分析的许多联系，这些联系并不推广到其他维度。三维物体，包括纽结、链环和三维流形（后者是从三维空间中粘出来的全局物体），承认组合不变量，也称为量子不变量，它们来自代数结构，也可以从二维共形场论中恢复。在大多数情况下，四维物体的量子不变量并没有组合描述，它们的定义和计算需要分析工具。我们希望通过构造新的四维组合不变量来弥合这一差距，其次，通过找到已知的四维流形分析不变量的组合描述，包括Donaldson-Floer和Seiberg-Witten不变量。

The project aims to develop and understand homological invariants of three- and four-dimensional topological objects. Such invariants include Donaldson-Floer and Seiberg-Witten theories, and several bigraded homology theories of links. These theories have the Alexander and Jones polynomials as their Euler characteristics, as well as the quantum sl(3) link invariant. We would like to construct a bigraded homology theory of links for each complex simple Lie algebra g, with components of links colored by irreducible representation of g. The Euler characteristic of the theory should be the quantum invariant of colored links associated with the quantum deformation of g, and the theory should be functorial (extend to cobordisms of links). Our other goals include better understanding of the relations between existing theories, and an investigation of the categories that appear when link homology theories are extended to tangles. Topological objects in dimensions three and four have special properties and a number of connections to algebra and analysis that do not generalize to other dimensions. Three-dimensional objects, including knots, links, and three-manifolds (the latter are global objects glued out of three-dimensional spaces), admit combinatorial invariants, also known as quantum invariants, that come from algebraic structures and can also be recovered from two-dimensional conformal field theories. Quantum invariants of four-dimensional objects, for the most part, are not known to have combinatorial descriptions, and their definition and computation requires analytical tools. We would like to bridge this gap by constructing new four-dimensional invariants that are combinatorial, and, second, by finding combinatorial description of known analytical invariants of four-manifolds, including Donaldson-Floer and Seiberg-Witten invariants.