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

该项目涉及将舒伯特演算的大量扩展扩展到仿射格拉斯曼尼亚人和仿射旗帜品种，称为‘’仿射舒伯特演算‘’。经典Schubert演算是计数代数几何的一个分支，它涉及计数满足某些交集条件的子空间，它是Hilbert第15个问题的解的结果。在现代公式中，舒伯特微积分通常被解释为齐次空间的上同调理论，最著名的是FLAG簇。仿射-舒伯特微积分的充分发展将解决麦克唐纳理论中的长期悬而未决的问题，并对物理问题产生影响，如Wess-Zumino-Witten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。仿射Schubert演算的新方法是由于最近发现了某些被称为k-Schur函数的明确定义的对称函数。在研究看似不相关的Macdonald理论时出现的k-Schur函数，最近被证明与仿射Grassmanian的几何和拓扑有关。我们将利用k-Schur函数的新组合来推导各种重数的公式，包括仿射Grassman流形和仿射标志流形中的交集重数。这些多样性中的一些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联系了来自组合学、几何学、表示论、物理学和计算的各种问题。将解决的主要问题可以从几个角度来看待：几何角度(例如“有多少条线满足一些一般的相交条件？”)、组合角度(“给定集合中有多少元素以及这些集合具有什么性质？”)、物理角度(“场如何关联？”)和计算方面(“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的跨学科。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。在项目结束时，还计划在菲尔兹学院为研究生举办暑期班。这项研究在很大程度上是由广泛的计算实验推动的。从该项目衍生的算法的健壮实现，将导致计算机代数系统的新包的开发。通过开放源码计算包传播这一新软件，不仅将推进拟议的研究计划，还将对数学、物理和计算机科学界产生广泛影响。

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.