Homological algebra and topology in three and four dimensions

该项目旨在发展和理解三维和四维拓扑对象的同调不变量。这类不变量包括Donaldson-Floer理论和Seiberg-Witten理论，以及若干连杆的梯度同调理论。这些理论以亚历山大多项式和琼斯多项式作为欧拉特征，以及量子sl(3)链接不变量。我们想要为每一个复简单李代数g构造一个连杆的重阶同调理论，其中连杆的分量由g的不可约表示着色。该理论的欧拉特征应该是与g的量子变形相关的有色连杆的量子不变量，并且该理论应该是泛函的（扩展到连杆的协数）。我们的其他目标包括更好地理解现有理论之间的关系，以及研究当链接同源理论扩展到缠结时出现的类别。维度3和维度4中的拓扑对象具有特殊的性质，并且与代数和分析有许多联系，这些联系不能推广到其他维度。三维物体，包括结、链和三流形（后者是粘出三维空间的全局物体），承认组合不变量，也称为量子不变量，来自代数结构，也可以从二维共形场论中恢复。四维物体的量子不变量在很大程度上没有组合描述，它们的定义和计算需要分析工具。我们希望通过构建新的组合四维不变量来弥合这一差距，其次，通过找到已知的四流形解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.