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

该项目涉及将舒伯特微积分扩展到仿射格拉斯曼函数和仿射旗簇，称为“仿射舒伯特微积分”。经典舒伯特微积分是枚举代数几何的一个分支，涉及计算满足某些交集条件的子空间，是解决希尔伯特第十五个问题的结果。在现代公式中，舒伯特微积分通常用齐次上同调理论来解释。仿射舒伯特微积分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，例如韦斯-祖米诺-维滕共形场理论模型的推广和本征函数为 k-舒尔函数的卡洛杰罗-萨瑟兰量子力学模型的扩展，仿射舒伯特微积分的新方法是通过最近发现的某些明确定义的对称函数而成为可能的。 k-Schur 函数是在看似无关的麦克唐纳理论的研究中出现的，最近被证明与仿射格拉斯曼函数的几何和拓扑有关，将利用 k-Schur 函数的新颖组合来推导各种重数的公式，包括仿射格拉斯曼和仿射旗流形中的交集重数，其中一些重数已知出现在麦克唐纳理论中。 WZW 模型的 Verlinde 融合系数。这个多方面的项目涉及并将组合学、几何、表示理论、物理学和计算的各种问题联系在一起。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“有多少条线满足多个通用相交条件？”）、组合角度（“给定集合中有多少元素以及这些集合有什么属性？”）、物理角度（“场如何存在”）。相关性？”），以及计算方面（“是否有有效的算法来计算这些数字或物体？”）。该项目是一个国际合作研究项目，核心小组成员分布在加拿大、美国、智利和法国，是跨学科的，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接受专业培训，并从与研究团队的互动中受益。项目结束时还计划在菲尔兹研究所为研究生举办暑期学校。调查主要是在广泛的计算实验的推动下，该项目的算法的稳健实施将导致计算机代数系统新软件包的开发。通过开源计算软件包传播这种新软件，不仅将推进拟议的研究计划，而且还将对数学、物理和计算机科学界产生影响。

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.