Combinatorial Representation Theory from Knot Theory and Algebraic Geometry

这个项目的核心是数据不完整的问题：我们只有部分信息，需要估计丢失的数据，或者只根据我们零碎的知识找到解决问题的方法。在这个项目中，我们使用了几何学和组合学的工具，即复杂的计数技术，使我们能够分析从DNA测序到优化的各种应用中的模式。这项工作大大有助于培养一支更加多样化的数学劳动力队伍，既增加了妇女和代表性不足的少数群体的渠道，又进一步加强了下游的保留。具体研究内容包括：1)组合分析不可约对称群表示的两个重要基网基和Tableau基之间的转移矩阵；2)描述Springer纤维和更一般的Hessenberg簇的分量的几何和拓扑；3)计算Hessenberg簇的(等变)上同调环；4)描述不同边标记图的广义样条线。PI是Hessenberg变种和等变上同源的GKM方法方面的专家；PI关于这些主题的一系列论文，加上原田、罗兹和其他人的新工作，使该领域处于新发现的边缘。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The core of this project is the problem of incomplete data: situations where we only have partial information and need either to estimate the missing data or to find ways to solve the problem based only on our fragmentary knowledge. In this project, we use tools from geometry as well as from combinatorics, namely the sophisticated counting techniques that allow us to analyze patterns in diverse applications from DNA sequencing to optimization. The work contributes significantly to developing a more diverse workforce in mathematics, both increasing the pipeline for women and underrepresented minorities and strengthening retention further downstream. This includes incorporating student researchers at all stages of their careers into the PI's lab.The specific research addressed in this proposal is: 1) combinatorially analyzing the transition matrices between two important bases of irreducible symmetric-group representations, the web basis and the tableau basis; 2) describing the geometry and topology of components of Springer fibers and, more generally, Hessenberg varieties; 3) computing the (equivariant) cohomology rings of Hessenberg varieties; and 4) describing generalized splines for different edge-labeled graphs. The PI is an expert in Hessenberg varieties and the GKM approach to equivariant cohomology; a sequence of the PI's papers on these subjects, together with new work by Harada, Rhoades, and others, leave the field poised on the brink of new discoveries.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.