Algebraic combinatorics and its application to algebraic geometry and low dimensional topology

自20世纪80年代以来，现代几何学在与数学物理学的深刻联系的推动下，经历了一个漫长而激烈的活动。 出于同样的原因，现代纽结理论也取得了进展，部分原因是数学物理和纽结理论中出现了一个特殊的基本方程，即杨-巴克斯特方程。随着研究在这些领域的进展，越来越明显的是，在关于单个结不变量的问题中，隐藏着关于瓦西里耶夫不变量的非常复杂的新问题，以及受复杂关系约束的新对象或图的集合。这些新问题本质上是组合性的，可以抽象地表达，而不涉及它们的原始上下文。 这种图的一个例子是量子场论中的费曼图。

Since about the 1980's there has been a long period of remarkable and intense activity in modern geometry, spurred on by the deep connexions with mathematical physics. For the same reason, there were advances in Modern Knot Theory, prompted partly by the occurrence of a particular fundamental equation, the Yang-Baxter Equation, in both mathematical physics and knot theory. As research progressed in these areas, it became increasingly apparent that deeply within questions about individual knot invariants lay new questions of great complexity about Vassiliev invariants, and aggregates of new objects, or diagrams, constrained by complex relations. These new questions are essentially combinatorial in nature, and may be expressed abstractly, without reference to their original context. An instance of such a diagram is the Feynman diagram from quantum field theory.