Combinatorics, Representations, and Catalan Theory

对称函数理论是一个数学研究领域，它产生了大量复杂的(有时是猜想的)恒等式，其中许多很难证明。对称函数与几何、表示理论和物理有联系，这些领域之间存在着深刻的相互作用：对称函数为这些其他领域中的困难问题提供了计算模型，而对称函数恒等式通常最好被理解为来自其他领域的更深层次概念的影子。这个项目研究被称为Delta猜想的对称函数问题的代数和几何基础。该项目为本科生和研究生提供了研究培训的机会。n元多项式环的商与排列的组合学和旗簇的几何有着经典的联系。这个项目研究了一个较新的环的类似商(和子空间)，这个环被称为‘超空间’，它有n个对易生成元(对应于玻色子)和n个反对易生成元(对应于费米子)。这些新的代数对象与有序集合划分的组合(如Haglund、Shimozono和PI所研究的)以及由Pawlowski和PI引入的各种生成线构形的几何相关联。Zabrocki的一个深度猜想，以及Wilson和PI的一个相关猜想，断言超空间(变体)给出了Delta猜想的一个表示论模型。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，这可能会导致对此的有启发性的证明。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.