V2EX schubert calculus

Schubert Calculus

定义 Definition

Schubert calculus(舒伯特演算/舒伯特微积分):代数几何与组合数学中的一个领域,主要研究旗形簇(flag varieties)与格拉斯曼流形(Grassmannians)上的Schubert 子簇的交理论与上同调环结构,用来解决诸如“满足若干几何条件的子空间有多少个”这类枚举几何问题。(在不同语境下也可泛指相关的组合规则,如 LittlewoodRichardson 规则。)

发音 Pronunciation (IPA)

/ubrt klkjls/

例句 Examples

Schubert calculus helps count how many lines satisfy certain geometric conditions.
舒伯特演算有助于计算满足特定几何条件的直线有多少条。

Using Schubert calculus, the intersection numbers of Schubert classes on a Grassmannian can be computed via combinatorial rules.
借助舒伯特演算,可以通过组合规则计算格拉斯曼流形上舒伯特上同调类的交数。

词源 Etymology

Schubert calculus 以德国数学家 Hermann Schubert(赫尔曼舒伯特,18481911)/strong>命名。他在 19 世纪末系统发展了枚举几何中的计数方法;后来这些思想被现代代数几何与上同调理论严格化,并与表示论、对称函数、组合学等方向深度交织,因此形成今天所说的“Schubert 演算”。

相关词 Related Words

文学与经典作品 Literary Works

  • William Fulton, Young Tableaux: With Applications to Representation Theory and Geometry(多处以 Schubert calculus 解释格拉斯曼流形与相关组合规则)
  • William Fulton, Intersection Theory(在交理论框架下讨论与 Schubert 类相关的计算思想与工具)
  • David A. Cox, John Little, Donal O’Shea, Ideals, Varieties, and Algorithms(在应用与入门语境中提及枚举几何/格拉斯曼相关问题时常涉及 Schubert calculus)
  • David Eisenbud, Joe Harris, 3264 and All That: A Second Course in Algebraic Geometry(以枚举几何为主线,出现 Schubert calculus 相关讨论)
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