Geometry and Asymptotics of Schubert Polynomials, Graph Colorings, and Flows on Graphs

受现实世界活动启发，数学中的三个基本和具有挑战性的问题是计算通过网络运输货物的方式的数量，对任务列表进行排序，以及将作业调度到时间段。这些计数问题的一些实例很难准确计数，人们可以研究这些数字的界、渐近性和大规模行为，以及使用编码被计数对象的几何结构。抽象地说，这些问题可以用下列数学对象来研究：“图上的整数流”、“置换的舒伯特多项式”和“图的顶点着色”，它们分别与其他数学领域和其他领域如计算机科学和物理有关。更具体地说，这个项目研究与表示理论和几何有关的计数、代数和渐近组合学中的问题。该项目包括三个部分。第一部分是关于寻找Zeilberger常数项恒等式的q-模拟来计算流动多面体的体积，并建立该恒等式与著名的Selberg积分之间的联系。第二部分利用已有的组合模型，如RC图、无碰白日梦和激发图，研究了Schubert多项式和Grothendieck多项式的大规模行为。最后一部分研究了著名的Stanley-Stembridge-Shareshian-Wachs型猜想的对象--Dyck路的色对称函数。PI将学习牛顿多面体、洛伦兹性质，以及与这些对称函数的Q-rook理论的联系。学生将在这个项目的过程中接受培训。这一奖项反映了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。

Three fundamental and challenging problems in mathematics inspired from real-world activities are to count the number of ways of transporting goods through a network, sorting a list of tasks, and scheduling jobs to time slots. Some instances of these counting problems are difficult to count exactly and one can instead study bounds, asymptotics, and large-scale behaviors of these numbers as well as using geometric structures encoding the objects that are counted. Abstractly these problems can be studied with the following mathematical objects: "integer flows on a graph", "Schubert polynomials of permutations", and "graph vertex colorings", respectively that have connections to other fields of mathematics and other areas like computer science, and physics.More concretely, this project studies problems in enumerative, algebraic, and asymptotic combinatorics with connections to representation theory and geometry. The project has three parts. The first part is about finding a ¨q-analogue¨ of a constant term identity of Zeilberger to compute volumes of flow polytopes and establishing a connection between this identity and the famous Selberg integral. The second part is about studying the large-scale behavior of Schubert and Grothendieck polynomials using existing combinatorial models like rc-graphs, bumpless pipe dreams, and excited diagrams. The last part is about studying chromatic symmetric functions of Dyck paths, which are the object of the famous Stanley--Stembridge--Shareshian--Wachs conjecture. The PI will study the Newton polytope, Lorentzian property, and connections to q-rook theory of these symmetric functions. Students will be trained during the course of this project.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.