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

本项目主要研究与Macdonald多项式相关的k-Schur函数理论、Gromov-Witten不变量、仿射Grassmannians的co（同调）和k理论以及共形场论的WZW模型中缺失的核心思想。k-Schur函数是在麦克唐纳多项式的研究中发现的对称函数。对k-舒尔函数的后续研究导致了一种理论，它沿着现代舒伯特微积分的路线分支到许多领域。经典舒伯特微积分解决了射影几何中的枚举问题，并利用交理论将其转化为格拉斯曼上同环中的计算问题。使用组合舒尔理论明确实现这种计算，在将舒伯特微积分转变为一种当代理论方面发挥了重要作用，该理论为各种各样的问题提供了优雅的解决方案。以类似的方式，k-Schur函数的引入已经在几何、表示理论、代数和物理中产生了一系列开放问题。组合学是一个广阔的数学领域，可以粗略地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起着不可或缺的作用，从生物学家到理论物理学家，许多科学家都使用组合方法。PI研究对称函数的组合学的改进，这是数学的一个经典部分，在物理、工程和计算机科学等领域都有应用。

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.