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

这个项目涉及舒伯特演算的一个巨大的扩展，仿射格拉斯曼和仿射旗品种，称为“仿射舒伯特演算”的发展。经典Schubert演算是计数代数几何的一个分支，研究满足一定交条件的子空间的计数问题，是希尔伯特第十五问题的结果。在现代表述中，舒伯特演算通常被解释为齐性空间的上同调理论，最著名的是旗簇。仿射Schubert微积分的全面发展将解决Macdonald理论中长期悬而未决的问题，并对Wess-Zumino-维滕共形场论模型的推广和本征函数为k-Schur函数的Calogero-Sutherland量子力学模型的扩展等物理问题产生影响。仿射舒伯特演算的新方法是最近发现的某些明确定义的对称函数称为k-舒尔函数。在看似无关的麦克唐纳理论的研究中出现的k-舒尔函数，最近被证明与仿射格拉斯曼的几何和拓扑有关。新的组合k-Schur函数将被用来推导公式的各种多重性，包括交叉多重性的仿射格拉斯曼和仿射旗流形。这些多重性中的一些已知出现在麦克唐纳理论中，并作为WZW模型的Verlinde融合系数。这个多方面的项目涉及并联系了组合学，几何学，表示论，物理学和计算的各种问题。主要的问题，将解决可以从几个角度来看：一个几何的角度（问题，如“有多少线满足一些通用的相交条件？“），组合的观点（“给定的集合中有多少元素，这些集合有什么性质？），物理学的观点（“场如何相互关联？”），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，跨学科，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，并将从与研究团队的互动中受益。还计划在项目结束时在菲尔兹研究所为研究生举办暑期学校。这项研究在很大程度上是由大量的计算实验推动的。该项目所产生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一新软件，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.