Symplectic Geometry and Schubert Calculus

DMS-0305128丽贝卡戈尔丁辛几何发生在组合学，表示论，物理学，几何学和拓扑学的十字路口。哈密顿T空间最重要的例子之一是紧李群的余伴随轨道。余伴轨道出现在物理学中，因为它们是所有具有特定谱的矩阵的集合。它们也出现在表示论中，因为复约化李群的所有不可约表示都出现在余伴轨道上的线丛的全纯截面中。它们在代数几何中作为射影簇出现。自然产生的子簇的交理论，称为舒伯特簇，出现在线性代数，赋值环和表示论的问题中。舒伯特演算是重要的，因为它是线性的情况下（因此第一步走向）理解复杂的代数交叉更普遍。在代数上，舒伯特演算分析了旗簇的上同调的乘法结构。主要的开放性问题是找到一个完全正的公式（等变）结构常数。在与A。克努森，戈尔丁已经发现了新的技术，使希望在某些情况下，积极的公式。戈尔丁还对拓扑不变量感兴趣，这些拓扑不变量可以区分任何紧致哈密顿T空间的约化空间，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。戈尔丁还开发了一些技术，利用少量关于环面如何作用于原始辛流形的信息来计算某些辛约化空间上的积分。理解舒伯特微积分是朝着如何研究代数簇的交叉这一更大问题迈出的一小步。这种交叉出现在数学之外，因为科学家提出的问题涉及几个物理约束的同时存在，并希望了解可能存在的解决方案。辛几何是描述物理系统的自然环境。例如，粒子的位置和动量的空间被称为“相空间”，它有一个自然的辛结构。在许多情况下，有很多对称性;例如，氢原子的物理观测不受观察者与粒子的固定距离的影响。“约化空间”的研究是一种研究物理学的方法，同时通过忽略（实际上是分割）这些对称性来简化空间。戈尔丁的大部分工作都涉及在这些简化系统上进行计算的技术，这可能会揭示物理学家和化学家对分子和粒子提出的问题。戈尔丁也在写一本书（与G。论数学教育中的神话。

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.