FRG: Collaborative Research: Affine Schubert Calculus: Combinatorial, geometric, physical, and computational aspects

这个项目涉及到将舒伯特演算扩展到仿射格拉斯曼和仿射旗的发展，称为“仿射舒伯特演算”。经典舒伯特微积分是希尔伯特第十五问题解的结果，它是数列代数几何的一个分支，研究满足一定交点条件的子空间的计数。在现代表述中，舒伯特演算通常在齐次空间的上同调理论中解释，最显著的是标志变分。仿射舒伯特演算的全面发展将解决麦克唐纳理论中长期存在的开放性问题，并对物理问题产生影响，如wuss - zumino - witten共形场论模型的推广和特征函数为k-Schur函数的Calogero-Sutherland量子力学模型的扩展。仿射舒伯特微积分的新方法是由最近发现的某些显式定义的对称函数k-舒尔函数实现的。k-舒尔函数，出现在看似无关的麦克唐纳理论的研究中，最近被证明与仿射格拉斯曼的几何和拓扑结构有关。将利用k-Schur函数的新组合来推导各种多重性的公式，包括仿射格拉斯曼和仿射标志流形中的交多重性。其中一些多重性已知出现在麦克唐纳理论和WZW模型的Verlinde融合系数中。这个多方面的项目涉及并结合了组合学、几何、表示理论、物理和计算的各种问题。可以从几个角度来看待将要解决的主要问题：几何角度（例如“有多少条线满足一些通用的相交条件？”）组合视角（“给定集合中有多少元素，这些集合有什么属性？”），从物理学的角度（“场是如何相互关联的？”）以及计算方面（“是否有有效的算法来计算这些数字或对象？”）。该项目是一个国际合作研究项目，核心小组成员位于加拿大、美国、智利和法国，涉及数学家、物理学家和计算机科学家。研究生将通过直接参与研究获得专业培训，并将从与研究团队的互动中受益。在项目结束时，还计划在菲尔兹研究所为研究生开设一所暑期学校。这项调查在很大程度上是由大量的计算实验推动的。从该项目中获得的算法的健壮实现将导致计算机代数系统的新软件包的开发。这个新软件的传播通过一个开源计算包，不仅将推进拟议的研究计划，而且将对数学，物理和计算机科学社区产生广泛的影响。

This project concerns the development of a vast extension of Schubert calculus to affine Grassmannians and affine flag varieties, called ``affine Schubert calculus". Classical Schubert calculus, a branch of enumerative algebraic geometry concerned with counting subspaces satisfying certain intersection conditions, is the outcome of the solution to Hilbert's Fifteenth problem. In the modern formulation, Schubert calculus is usually interpreted in cohomology theories of homogeneous spaces, most notably flag varieties. The full development of affine Schubert calculus will solve long-standing open problems in Macdonald theory and have an impact on physical questions, such as generalizations of Wess-Zumino-Witten conformal field theory models and extensions of Calogero-Sutherland quantum mechanical models whose eigenfunctions are k-Schur functions. The new approach to affine Schubert calculus is made possible by the recent discovery of certain explicitly defined symmetric functions called k-Schur functions. The k-Schur functions, which arose in the study of the seemingly unrelated Macdonald theory, were recently shown to be connected to the geometry and topology of the affine Grassmanian. The novel combinatorics of k-Schur functions will be exploited to deduce formulae for various multiplicities, including intersection multiplicities in the affine Grassmannian and the affine flag manifold. Some of these multiplicities are known to occur in Macdonald theory and as Verlinde fusion coefficients for the WZW model.This many-faceted project involves and ties together various problems from combinatorics, geometry, representation theory, physics, and computation. The main questions that will be addressed can be viewed from several points of view: a geometric perspective (questions such as "how many lines are there satisfying a number of generic intersection conditions?"), a combinatorial perspective ("how many elements are in given sets and what properties do these sets have?"), a physics perspective ("how do fields correlate?"), and computational aspects ("are there efficient algorithms for calculating these numbers or objects?"). The project is an international cooperative research venture, with core group members located in Canada, the United States, Chile, and France, and interdisciplinary, involving mathematicians, physicists, and computer scientists. Graduate students will receive professional training by direct involvement in the research and will benefit from interaction with the research team. A summer school at the Fields institute for graduate students is also planned at the conclusion of the project. 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. The dissemination of this new software through an open-source computational package, will not only advance the proposed research program but will also have an outreach impact on the mathematics, physics, and computer science communities.