Combinatorial Methods in Geometric Representation Theory

表征理论家试图量化、分析和预测一组运动（如旋转或反射）如何作用于物理和几何对象。 几何表示理论的基本问题是使用几何工具构造表示。组合表示论的基本问题是用基本的组合参数来明确地理解表示：例如，将给定的表示分解为不可约表示的基本构建块。这两种表征理论家都试图理解表征，但几何表征理论家似乎回答了一个问题，比如“找到一个自然的表征”，而组合表征理论家只回答了部分问题，他们试图“明确地分析一个给定的表征”。这个建议解决了几何和组合表示理论中的几个问题，使用组合学，代数几何和拓扑学的工具。 更广泛的影响包括许多其他数学领域的工作，包括纽结理论和交换代数，以及数学物理等领域。 在教育方面，PI将在研究和教学方面建立研究生保留和卓越的计划。

Representation theorists seek to quantify, analyze, and predict how a group of motions---like rotations or reflections---act on physical and geometric objects. The fundamental problem in geometric representation theory is to construct representations using geometric tools. The fundamental problem in combinatorial representation theory is to understand a representation explicitly in terms of essential combinatorial parameters: for instance to decompose a given representation into the fundamental building blocks of irreducible representations. Both kinds of representation theorists seek to understand representations, but a problem that may seem answered to geometric representation theorists, such as `find a natural representation', is only partially answered to combinatorial representation theorists, who seek to `analyze explicitly a given representation'. This proposal addresses several problems in geometric and combinatorial representation theory, using tools from combinatorics, algebraic geometry, and topology. The broader impact includes work in many other areas of mathematics, including knot theory and commutative algebra, as well as fields like mathematical physics. Educationally, the PI will establish programs for graduate student retention and excellence, in both research and teaching.