Knot Theory, Algebra, and Higher Algebra

纽结理论、代数和高等代数

• 批准号: 262178-2013
• 负责人: BarNatan, Dror
• 金额: $ 2.11万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635484
• 关键词: Knot; Theory; Algebra; Higher

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, athttp://www.math.toronto.edu/~drorbn/.

为什么数学家对整数如此着迷？当然不是因为盯着9465438这样的数字所固有的美。也不是因为很难算出9，465，438是2x 3x 1，577，573。真正的原因是，整数是令人惊讶的深度，和研究整数，也被称为“数论”，迫使我们更好地理解，并确实发展，许多其他有用的和美丽的技术，概念和想法。数论似乎与一切事物都有联系。同样地，尽管在较小的范围内，许多像我这样的纽结理论家对鞋带并不关心，但却非常关心打结鞋带的研究与三维流形、双曲几何、量子场论、微分几何、李理论和表示论等先验的遥远学科之间错综复杂、深刻联系的意想不到的方式，量子代数，组合数学，同调代数和复杂的算法。我对这个项目的研究将集中在进一步阐述这些意想不到的联系，使用分析和计算工具。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，其中结变得更像代数工具，而不是拓扑（但某些类别的虚结描述了四维空间中的某些类别的结）。“同态展开式”是一类与代数和量子场论有着深刻联系的纽结不变量，而“虚纽结”与“同态展开式”的结合可能会为所有量子群的研究提供一个统一的框架，从而使代数学受益。我倾向于写论文，做讲座，画画，写计算机程序。因此，我在这个项目中的大部分工作将最终找到我已经全面的网站，athttp：www.math.toronto.edu/~drorbn/。