Symplectic Geometry and Schubert Calculus

辛几何和舒伯特微积分

• 批准号: 0305128
• 负责人: Rebecca Goldin
• 金额: $ 7.86万
• 依托单位: George Mason University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-15 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305128&HistoricalAwards=false
• 关键词: Symplectic; Geometry; Schubert; Calculus

DMS-0305128Rebecca GoldinSymplectic geometry occurs at a crossroad of combinatorics, representation theory, physics, geometry and topology. One of the most important examples of a Hamiltonian T-space is the coadjoint orbit of a compact Lie group. Coadjoint orbits arise in physics, as they are the set of all matrices with specified spectra. They also occur in representation theory, as all irreducible representations of a complex reductive Lie group occur as holomorphic sections of a line bundle over a coadjoint orbit. They occur in algebraic geometry as projective varieties. The intersection theory of naturally arising subvarieties, called Schubert varieties, appears in questions of linear algebra, valuation rings, and representation theory. Schubert calculus is important as it is the linear case for (and hence the first step towards) understanding complex algebraic intersections more generally. Algebraically, Schubert calculus analyses the multiplicative structure of the cohomology of flag varieties. The main open question is to find a totally positive formula for the (equivariant) structure constants. In joint work with A. Knutson, Goldin has discovered new techniques that give hope for a positive formula in certain cases. Goldin is also interested in topological invariants that distinguish reduced spaces for any compact Hamiltonian T-space, such as certain ideals found in its equivariant cohomology ring or generalized equivariant Euler classes. Goldin is additionally developing techniques to calculate integrals on certain symplectic reduced spaces using small amounts of information about how the torus acts on the original symplectic manifold.Understanding Schubert calculus is a small step towards the larger question of how to study intersections of algebraic varieties. Such intersections arise outside of mathematics, as scientists ask questions that involve the concurrence of several physical constraints, and want to understand what possible solutions there may be. Symplectic geometry is a natural setting to describe physical systems. The space of the position and the momentum of a particle, for example, is called "phase space" and has a natural symplectic structure on it. In many cases, there is a lot of symmetry; physical observations of the Hydrogen atom, for example, are unaffected by the position of the observer a fixed distance away from the particle. The study of "reduced spaces" is a way of studying the physics while simplifying the space by ignoring (in fact, dividing out) these symmetries. Much of Goldin's work involves techniques to do computations on these simplified systems, which may shed light on questions that physicists and chemists are posing about molecules and particles. Goldin is also writing a book (joint with G. Goldin) on myths in mathematics education.

辛几何是组合学、表示理论、物理学、几何学和拓扑学的交叉学科。哈密顿t空间的一个最重要的例子是紧李群的协伴轨道。伴随轨道在物理学中出现，因为它们是具有特定光谱的所有矩阵的集合。它们也出现在表示理论中，因为复约李群的所有不可约表示都出现在共伴轨道上的线束的全纯截面上。它们在代数几何中作为射影变体出现。自然产生的子变种的交点理论，称为舒伯特变种，出现在线性代数、估值环和表示理论的问题中。舒伯特微积分很重要，因为它是线性的情况（因此是第一步）更普遍地理解复杂的代数交集。在代数上，舒伯特演算分析了旗变体上同调的乘法结构。主要的开放问题是为（等变）结构常数找到一个完全正的公式。在与a .克努森的合作中，戈尔丁发现了一些新技术，这些技术给某些情况下的正公式带来了希望。Goldin还对区分任何紧哈密顿t空间的约化空间的拓扑不变量感兴趣，例如在其等变上同调环或广义等变欧拉类中发现的某些理想。Goldin还在开发技术，利用环面如何作用于原始辛流形的少量信息来计算某些辛简化空间上的积分。理解舒伯特微积分是朝着如何研究代数变量的交集这一更大问题迈出的一小步。这样的交叉出现在数学之外，因为科学家提出的问题涉及到几个物理约束的并发，并希望了解可能的解决方案。辛几何是描述物理系统的一种自然设置。例如，粒子的位置和动量的空间被称为“相空间”，它具有自然的辛结构。在许多情况下，有很多对称；例如，对氢原子的物理观测不受观测者距离粒子一定距离的位置的影响。“简化空间”的研究是一种研究物理的方法，同时通过忽略（实际上是划分）这些对称性来简化空间。戈尔丁的大部分工作涉及对这些简化系统进行计算的技术，这可能会为物理学家和化学家提出的关于分子和粒子的问题提供启发。戈尔丁还在写一本关于数学教育中的神话的书（与G.戈尔丁合作）。