Homological Algebra Methods in Topology and Combinatorics

在过去的二十年里，低维拓扑对两类不变量进行了大量的研究：四维的规范理论不变量和三维的组合不变量。虽然两者都与物理学有着深厚的联系，但它们几乎没有共同的技术，而且有相当不同的风格。然而，随着Khovanov和Ozsvath-Szabo最近的研究，这种情况可能会改变。1999年，Khovanov引入了结点的梯度同调理论，并证明了它的梯度欧拉特征是琼斯多项式。这被证明是琼斯多项式的一个深远推广。此外，有强有力的证据表明，Khovanov理论和Ozsvath-Szabo理论可以在规范理论类型和组合类型不变量之间架起桥梁。在Khovanov工作的激励下，PI和他的学生Laure Helme-Guizon建立了图的梯度同调理论，该理论在取欧拉特征时产生色多项式。PI打算进一步研究这些同调理论，包括结和图。一些具体问题是：理解它们的几何意义，研究它们在各种剪切和粘贴操作下的行为，为结和图的各种其他多项式构造同调理论，以及研究低维拓扑中与其他不变量的关系。低维拓扑研究三维和四维空间的形状。由于我们的空间和时空的维度，这些维度对人类具有特殊的利益。结理论是低维拓扑中的一个子领域，它研究三维空间中的结性。结是值得研究的，不仅因为它们是三维空间结构的基础，而且因为它与数学以外的领域有联系。例如，生化学家已经发现了打结的DNA分子（1980年代）和打结的蛋白质（2004年）。它还与图论的研究密切相关，图论是数学家、计算机科学家和其他人感兴趣的领域。在过去的二十年里，在规范理论、量子代数和数学物理的推动下，低维拓扑中出现了大量新的不变量。特别是，由Khovanov利用同调代数的思想提出的一个新的结点不变量，最近引起了人们的极大兴趣。此后，PI和他的学生提出了类似的图论。本项目旨在研究这些新的不变量，特别强调结点和图的同调代数方法。

Over the past two decades, low dimensional topology has seen a greatdeal of studies in two types of invariants: gauge theory type invariants in dimension four and combinatorial type invariants in dimension three. While both sides have deep connections with physics, they share little common techniques and have rather different flavors. This picture could change though, with recent work due to Khovanov and Ozsvath-Szabo.In 1999, Khovanov introduced a graded homology theory for knots,and proved that its graded Euler characteristic is the Jones polynomial.This has turned out to be a far reaching generalization of the Jones polynomial. Furthermore, there is strong evidence that Khovanov theory, along with the Ozsvath-Szabo theory, could bridge the connection between gauge theory type and combinatorial type invariants.Motivated by Khovanov's work, the PI, with his student Laure Helme-Guizon, has established a graded homology theory for graphs which yields the chromatic polynomial when taking Euler characteristic. The PI intends to further his investigation on these homology theories, both for knots and for graphs. Some of the specific problems are: understanding their geometric meanings, studying their behavior under various cut and paste operations, constructing homology theories for various other polynomials of knots and graphs, and investigating relations with other invariants in low dimensional topology.Low dimensional topology studies the shapes of three and four dimensional spaces. These dimensions are of particular interests to mankind because of the dimensions of our space and our space-time. A specific subfield in low dimensional topology is knot theory, which studies the knottedness in our three dimensional space. Knots are worthwhile to study not only because they are fundamental in 3-dimensional spatial structure, but also because of its connection to areas outside mathematics. For example, biochemists have discovered knotted DNA molecule (1980s) and knotted proteins (2004). It is also intimately related to the study of graph theory, an area interesting to mathematicians, computer scientists, and others. Over the past two decades, there have been a flourish of new invariants in low dimensional topology, boosted by ideas from gauge theory, quantum algebras, and mathematical physics. In particular, a new invariant for knots, developed by Khovanov using ideas in homological algebra, has sparked a great deal of interest recently. An analogous theory for graphs has since been developed by the PI and his student. This project aims to investigate these new invariants, with a particular emphasis on the homological algebra methods for knots and graphs.