Geometry, Arithmetic, and Combinatorics of Schubert Calculus

舒伯特微积分的几何、算术和组合学

• 批准号: 1303352
• 负责人: Harry Tamvakis
• 金额: $ 16.19万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303352&HistoricalAwards=false
• 关键词: Geometry; Arithmetic; Combinatorics; Schubert; Calculus

The investigator will study the generalization of the classical Schubert calculus to the homogeneous spaces of Lie groups and related applications to enumerative algebraic geometry, arithmetic geometry, quantum cohomology, and algebraic combinatorics. The proposed research problems include the following: (a) Formulate and prove a rule for computing the product of two Schubert classes in the cohomology of flag manifolds; (b) establish a Pieri type rule in the cohomology and quantum cohomology ring of homogeneous spaces; (c) find a good theory of quantum Schubert polynomials for the orthogonal and symplectic Lie groups; (d) study the combinatorial theory of theta and eta polynomials, in particular their double versions, with applications to equivariant cohomology; (e) achieve more efficient computations in the Arakelov geometry of flag varieties. The educational activities proposed by the investigator include incorporating problems stemming from his research into the University of Maryland High School Mathematics Competition. He also is working on a book on Schubert calculus and its various modern extensions which will emphasize the geometric aspects of the theory, in a uniform manner across the different Lie types.In the late 19th century, Hermann Schubert made a first systematic study of enumerative projective geometry and invented a calculus which enabled him to solve a plethora of enumerative problems in a systematic fashion. An example of the kind of question he addressed is: given four randomly generated lines in Euclidean 3-space, how many lines will intersect all four of the given lines? Schubert's fundamental work was a precursor of the cohomology and intersection theories of the 20th century, and their modern extensions today, with applications to quantum physics (via string theory) and number theory (via arithmetic intersection theory). Moreover, the entire story can be placed in a more symmetric framework, and told using the language of Lie groups and their representation theory, at an increasing level of combinatorial complexity and importance. The investigator has studied these theories over a period of many years and has shown how to achieve combinatorially explicit computations and formulas which give a global picture of the ring structure in each case. Recently, he has succeeded in doing this in a uniform way across all classical Lie types -- in the case of cohomology -- and has thus opened new avenues of research in the rapidly growing field of modern Schubert calculus.This award is co-funded by the Combinatorics Program.

调查员将研究经典舒伯特演算的推广到李群的齐次空间和相关应用枚举代数几何，算术几何，量子上同调和代数组合学。提出的研究问题包括：（a）在旗流形的上同调中建立并证明一个计算两个Schubert类乘积的定则;（B）在齐性空间的上同调环和量子上同调环中建立一个Pieri型定则;（c）在正交和辛李群中找到一个好的量子Schubert多项式理论;（d）研究theta和eta多项式的组合理论，特别是它们的双重形式，并将其应用于等变上同调;（e）在旗簇的Arakelov几何中实现更有效的计算。调查员提出的教育活动包括将他的研究中产生的问题纳入马里兰州大学高中数学竞赛。他也正在写一本书舒伯特演算及其各种现代扩展，这将强调几何方面的理论，在统一的方式在不同的李类型。在19世纪后期，赫尔曼舒伯特作出了第一个系统的研究枚举射影几何和发明了演算，使他能够解决过多的枚举问题，在一个系统的方式。一个例子的那种问题，他解决的是：给定四个随机生成的线在欧几里德3空间，有多少线将相交所有四个给定的线？舒伯特的基础工作是世纪上同调和交集理论的先驱，以及它们今天的现代扩展，应用于量子物理学（通过弦理论）和数论（通过算术交集理论）。此外，整个故事可以放在一个更对称的框架中，并使用李群及其表示理论的语言来讲述，组合的复杂性和重要性越来越高。调查研究了这些理论在一段时间的许多年，并已表明如何实现组合明确的计算和公式，使全球图片的环结构在每种情况下。最近，他成功地在所有经典李类型中以统一的方式做到了这一点-在上同调的情况下-因此在快速发展的现代舒伯特微积分领域开辟了新的研究途径。该奖项由组合数学计划共同资助。