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CAREER: Interactions between knot homology and rep

职业：结同源性和重复之间的相互作用

基本信息

批准号：
1255334
负责人：
Aaron Lauda
金额：
$ 44.5万
依托单位：
University of Southern California
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2013
资助国家：
美国
起止时间：
2013-07-01 至 2019-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1255334&HistoricalAwards=false
关键词：
CAREER Interactions between knot homology

项目摘要

This proposal details a comprehensive program which uses diagrammatic methods in categorified representation theory to obtain new results in representation theory and low-dimensional topology. One research objective seeks to exploit an analogy between knot homology and categorified representation theory to produce new representation theoretic objects arising from odd Khovanov homology. Another research direction seeks to use the powerful tool of categorical skew Howe duality to solve open problems in link homology theory for the special linear Lie algebra. This work will have applications to Landau-Ginzburg models in theoretical physics via the Kapustin-Li formula. The same diagrammatics used to encode the structure of categorified quantum groups will be used to transform current research into unique educational opportunities including an undergraduate research program at USC and innovative curriculum development of an undergraduate course on 'Diagrammatic Algebra'. The PI will also tap into the existing infrastructure of USC's community involvement to recruit high school students for a mathematics based event using diagrammatic algebra as a tool for engaging high school students.An emerging mathematical philosophy known as "categorification" has significantly altered our perception of mathematics and the way we analyze our surroundings. Imagine if an architect studied a building by only examining its shadow. In much the same way, the perspective of categorification uncovers a hidden layer in mathematical objects allowing mathematicians to see the entire structure rather than its shadow. The PI's research uncovers these hidden structures in an area of mathematics known as representation theory which is closely connected to some of the most sophisticated models in theoretical physics. Following the philosophy that fundamental structures in mathematics should be simple and intuitive the PI utilizes a diagrammatic framework to encode a great deal of complexity into an intuitive diagrammatic language that greatly simplifies computations. The PI will incorporate this diagrammatic philosophy into transformative educational programs aimed at high school students, undergraduates, and graduate students making the tools of modern research accessible to a new generation of researchers. This award is co-funded by the Algebra and Number Theory program and the Topology and Geometric Analysis program.

该建议详细介绍了一个全面的计划，它使用图解方法在分类表示理论，以获得新的结果表示理论和低维拓扑。一个研究目标是利用纽结同调和范畴化表示理论之间的类比，产生新的表示理论对象所产生的奇霍瓦诺夫同调。另一个研究方向是利用范畴斜Howe对偶这一强有力的工具来解决特殊线性李代数的链同调理论中的公开问题。这项工作将通过Kapustin-Li公式应用于理论物理中的Landau-Ginzburg模型。用于对分类量子群的结构进行编码的相同数学将用于将当前的研究转化为独特的教育机会，包括南加州大学的本科研究项目和“图解代数”本科课程的创新课程开发。 PI还将利用南加州大学现有的社区参与基础设施，招募高中生参加一个基于数学的活动，使用图解代数作为吸引高中生的工具。一种新兴的数学哲学被称为"分类"，它极大地改变了我们对数学的看法和我们分析周围环境的方式。想象一下，如果一个建筑师只通过观察建筑物的影子来研究它。同样，分类的视角揭示了数学对象中隐藏的一层，使数学家能够看到整个结构，而不是它的影子。 PI的研究揭示了这些隐藏在数学领域中的结构，称为表征理论，它与理论物理中一些最复杂的模型密切相关。遵循数学的基本结构应该简单直观的哲学，PI利用图表框架将大量复杂性编码为直观的图表语言，大大简化了计算。 PI将把这种图解哲学纳入针对高中生，本科生和研究生的变革性教育计划，使新一代研究人员能够使用现代研究工具。该奖项由代数和数论计划以及拓扑和几何分析计划共同资助。