Combinatorics, Representations, and Catalan Theory

对称函数论是数学研究的一个领域，它产生了大量复杂的（有时是抽象的）恒等式，其中许多恒等式很难证明。对称函数与几何学、表示论和物理学都有联系，并且这些领域之间有着深刻的相互作用：对称函数为这些领域中的困难问题提供了一个计算模型，而对称函数恒等式通常最好理解为其他领域更深层次概念的影子。这个项目研究对称函数问题的代数和几何基础，称为Delta猜想。该项目为本科生和研究生提供了研究培训机会。n元多项式环的商与排列组合学和标志簇的几何学有着经典的联系。这个项目研究了一个新的环的类似的子空间（和子空间），称为“超空间”，它有n个交换生成元（对应于玻色子）和n个反交换生成元（对应于费米子）。这些新的代数对象被链接到组合学的有序集分区（研究由Haglund，Shimozono，和PI）和几何形状的各种跨越线配置介绍的Pawlowski和PI。Zabrocki的一个深层猜想，以及威尔逊和PI的一个相关猜想，断言超空间（的一个变体）给出了Delta猜想的表示论模型。PI将使用超空间的代数和几何将其他数学领域与Delta猜想联系起来，这可能会导致其启发性的证明。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。

Symmetric function theory is an area of mathematical research which has produced an abundance of intricate (and sometimes conjectural) identities, many of which are very difficult to prove. Symmetric functions have ties to geometry, representation theory, and physics, and there is a deep interplay between these areas: symmetric functions give a computational model for difficult matters in these other fields, and symmetric function identities are often best understood as shadows of deeper concepts from other areas. This project studies the algebraic and geometric underpinnings of a symmetric function problem called the Delta Conjecture. The project provides research training opportunities for undergraduate and graduate students.Quotients of the polynomial ring in n variables have classical ties to the combinatorics of permutations and the geometry of the flag variety. This project studies analogous quotients (and subspaces) of a newer ring called 'superspace' which has n commuting generators (corresponding to bosons) and n anticommuting generators (corresponding to fermions). These new algebraic objects are linked to the combinatorics of ordered set partitions (as studied by Haglund, Shimozono, and the PI) and the geometry of a variety of spanning line configurations introduced by Pawlowski and the PI. A deep conjecture of Zabrocki, and a related conjecture of Wilson and the PI, assert that (a variant of) superspace gives a representation-theoretic model for the Delta Conjecture. The PI will use the algebra and geometry of superspace to tie other areas of mathematics to the Delta Conjecture, which could lead to an illuminating proof thereof.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.