Knot Theory, Algebra, and Higher Algebra

为什么数学家们对整个数字着迷？当然不是因为盯着9,465,438这样的数字所固有的美。这也不是因为很难计算出9,465,438是2x3x1,577,573。真正的原因是，整数的深度出奇地深，而对整数的研究，也被称为“数论”，迫使我们更好地理解，甚至发展许多其他有用和美丽的技术、概念和思想。数论似乎与一切事物都有关。 同样，尽管规模较小，但许多结理论家，如我自己，对鞋带不太关心，但非常关心打结鞋带的研究如何以意想不到的方式与诸如三维流形、双曲几何、量子场论、微分几何、李理论和表象理论、量子代数、组合学、同调代数和复杂算法等先验的遥远学科复杂而深刻地联系在一起。 我对这个项目的研究将集中在使用分析和计算工具进一步阐述这些意想不到的联系上。更具体地说，我的主要目标将是完成我们对经典和“虚拟”结的“同态展开”的理解。“虚结”是一种特定类型的结理论的名称，在这种理论中，结变得更像是代数小玩意，而不是拓扑学的(然而，某些类别的虚拟结描述了4维空间中的某些类别的结)。“同态展开”是一类与代数和量子场论有着深厚联系的纽结不变量，而“虚拟纽结”和“同态展开”的结合可能会为所有量子群的研究提供一个统一的框架，从而使代数受益。 我倾向于写说明文、做说明性演讲、画图画和编写计算机程序。因此，我在这个项目中的大部分工作将最终进入我已经很全面的网站，网址是 Http://www.math.toronto.edu/~drorbn/.

Why are mathematicians fascinated by the whole numbers? Certainly not because of the beauty inherent in staring at numbers such as 9,465,438. Neither is it due to the difficulty in figuring out that 9,465,438 is 2x3x1,577,573. The true reason is that the whole numbers are surprisingly deep, and the study of whole numbers, also known as "number theory", forces us to better understand, and indeed develop, many other useful and beautiful techniques, concepts and ideas. Number theory just seems to be related to everything. Likewise, though on a smaller scale, many knot theorists such as myself care little about shoelaces, yet care a lot about the unexpected ways by which the study of knotted shoelaces is intricately and deeply related to such a priori remote subjects as 3-dimensional manifolds, hyperbolic geometry, quantum field theory, differential geometry, Lie theory and representation theory, quantum algebra, combinatorics, homological algebra and sophisticated algorithmics. My research for this project will concentrate on the further elaboration of these unexpected links, using both analytical and computational tools. More specifically, my primary goal will be to complete our understanding of "homomorphic expansions" of classical and "virtual" knots. "Virtual knots" is a name for a certain type of knot theory in which the knots become much more of algebraic gadgets, rather than topological (yet certain classes of virtual knots describe certain classes of knots in 4-dimensional space). "Homomorphic expansions" are a certain class of knot invariants with deep connections to algebra and to quantum field theory, and the marriage of "virtual knots" to "homomorphic expansions" will likely benefit algebra by providing a unified framework for the study of all quantum groups. I tend to write expositions and give expository talks, draw pictures and write computer programs. Thus much of my work in this project will end up finding its way to my already-comprehensive web site, at http://www.math.toronto.edu/~drorbn/.