Affine Schubert Calculus

这个项目研究了代数几何中出现的组合结构，特别是舒伯特微积分，它起源于19世纪末的计数几何，但其应用涉及到现代数学，如量子上同调和量子K理论。这项研究将集中在被称为仿射Grassmannians和仿射旗簇的无限维齐性空间，将使用组合、代数和几何方法来研究它们。特别是，对于这些变种的各种上同调环，将开发出显式计算方法。一个例子是利用Groebner基方法得到了一种特殊的线性群k-Schur函数的仿射Grassmanian同调基的新公式，它将Schur函数的经典表达式推广为Young Tableaux的权生成函数。这个项目应用于麦克唐纳多项式理论，并可能为旗形的Gromov-Witten不变量提供新的组合公式。这一研究在很大程度上是由大量的计算实验推动的。从该项目得出的算法的健壮实现将导致为计算机代数系统开发新的包，这些包允许用户构建和操纵数学对象的计算机实现。通过开放源码系统传播这一软件，不仅将推进拟议的研究计划，还将对数学、物理和计算机科学界产生广泛影响。

This project investigates combinatorial structures arising in algebraic geometry, particularly the Schubert calculus, whose origins are in the enumerative geometry of the late 19th century, but whose applications touch on modern mathematics such as quantum cohomology and quantum K-theory. The research will focus on the infinite-dimensional homogeneous spaces known as affine Grassmannians and affine flag varieties, which will be studied using combinatorial, algebraic, and geometric methods. In particular, explicit computational methods will be developed for various kinds of cohomology rings of these varieties. An example is the use of Groebner basis methods to obtain new formulae for the homology basis of the affine Grassmannian of a special linear group called k-Schur functions, which generalizes the classical expression of a Schur function as the weight generating function of Young tableaux. This project has applications to the theory of Macdonald polynomials and may provide new combinatorial formulae for Gromov-Witten invariants of flag manifolds.The investigation is largely fueled by extensive computational experimentation. The robust implementation of algorithms derived from the project will lead to the development of new packages for computer algebra systems, which allow a user to build and manipulate computer implementations of mathematical objects. The dissemination of this software, through an open-source system, will not only advance the proposed research program, but will also have an outreach impact on the mathematics, physics, and computer science communities.