Combinatorics of affine Schubert calculus, K-theory, and Macdonald polynomials

这个项目涉及Macdonald多项式的k-Schur函数理论、Gromov-Witten不变量、仿射Grassmannians的上(同调)和K-理论以及共形场理论的WZW模型中缺失的核心思想的研究。K-Schur函数是在研究Macdonald多项式时发现的对称函数。根据对k-Schur函数的研究，产生了一种沿着现代Schubert微积分的路线分支到许多领域的理论。经典的Schubert微积分解决了射影几何中的计数问题，并利用交理论将它们转化为Grassman上同调环上的计算问题。使用组合舒尔理论显式实现这种计算，在将舒伯特微积分转化为当代理论的过程中发挥了重要作用，该理论为各种问题提供了优雅的解决方案。以类似的方式，k-Schur函数的引入在几何、表示理论、代数和物理中产生了一系列悬而未决的问题。组合学是一个广阔的数学领域，大致被描述为对满足特定标准的对象集合进行计数的研究。因此，组合学在许多领域的发展中发挥着不可或缺的作用，从生物学家到理论物理学家的科学家都在使用组合方法。PI研究对称函数的组合精化，对称函数是数学的经典部分，在物理、工程和计算机科学等领域有应用。

This project concerns the study of missing core ideas in the theory of k-Schur functions connected to Macdonald polynomials, Gromov-Witten invariants, the co(homology) and K-theory of affine Grassmannians, and the WZW model of conformal field theories. The k-Schur functions are symmetric functions discovered in a study of Macdonald polynomials. Pursuant work with k-Schur functions has led to a theory that branches into many fields along the lines of modern Schubert calculus. Classical Schubert calculus addressed enumerative problems in projective geometry and used intersection theory to convert them into problems of computation in the cohomology ring of the Grassmannian. The explicit realization of such computations using combinatorial Schur theory played a major role in transforming Schubert calculus into a contemporary theory that provides elegant solutions to a diverse body of problems. In a similar manner, the introduction of k-Schur functions has generated a pool of open problems in geometry, representation theory, algebra, and physics. Combinatorics is a vast area of mathematics loosely described as the study of counting collections of objects that satisfy specified criteria. As such, combinatorics plays an integral role in the development of many fields, and combinatorial methods are employed by scientists ranging from biologists to theoretical physicists. The PI studies a refinement for the combinatorics of symmetric functions, a classical part of mathematics with applications in fields including physics, engineering, and computer science.