CAREER: Interactions between knot homology and rep

这项建议详细说明了一个全面的程序，它使用范畴化表示理论中的图示方法来获得表示理论和低维拓扑的新结果。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来产生由奇Khovanov同调产生的新的表示理论对象。另一个研究方向是利用范畴斜豪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。用于编码分类量子群结构的相同图解将被用于将当前的研究转化为独特的教育机会，包括在南加州大学的本科生研究计划，以及本科生关于“图解代数”的创新课程开发。PI还将利用南加州大学社区参与的现有基础设施，招募高中生参加一个以数学为基础的活动，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，即所谓的“归类”，显著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，如果一位建筑师只通过检查阴影来研究一座建筑。同样，分类的观点揭示了数学对象中的一个隐藏层，使数学家能够看到整个结构，而不是它的影子。PI的研究揭示了这些被称为表示理论的数学领域的隐藏结构，该理论与理论物理中的一些最复杂的模型密切相关。遵循数学的基本结构应该是简单和直观的哲学，PI利用图解框架将大量的复杂性编码成直观的图解语言，大大简化了计算。PI将把这一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获得现代研究的工具。该奖项由代数和数论项目以及拓扑学和几何分析项目共同资助。

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.