The Positive Grassmannian: Applications and Generalizations

这个项目探索位于组合学、表示论、统计物理学和可积系统的交叉点上的结构，对几个领域具有潜在的重大影响，包括应用于浅水波，蛋白质合成中的平移，以及超对称杨-米尔斯理论中的散射幅度。该项目数学中的中心结构是Grassmannians，它将向量空间的子空间参数化，在数学中普遍存在，在射影几何中以射影空间的形式出现，在微分几何中以紧致光滑流形的形式出现，在代数几何中以一种方案的形式出现。Grassmannians在计算机视觉的空间识别、编码和通信理论、物理学中的浅水波研究以及亚原子粒子散射幅度的计算中都发挥着重要的作用。这个项目探索了真正的格拉斯曼尼亚人中非常丰富的子集的特征，这些子集被称为完全积极和完全非负的格拉斯曼尼亚人。这些结构是对经典的正定和半正定矩阵理论以及在李论背景下的表示理论工作的改进和推广，将从拓扑学、表示论和组合的角度研究它们的结构。该项目还试图通过杰出女性在加州大学伯克利分校的一系列讲座来提高女性数学家的知名度。更具体地说，这个项目涉及围绕正Grassman多项式、Macdonald-Koornwinder多项式和非对称排斥过程的几个相互关联的问题。具体地说，这项研究将研究：旗簇镜像对称性的一个新的多面体表示；正Grassman的拓扑结构；来自Grassman的Kadomtsev？Petviashvili方程的孤子解的结构；正Grassman的新推广--幅面体的组合学；以及Macdonald-Koornder多项式的组合公式和非对称简单排除过程概率的菱形表格。

This project explores structures that lie at the intersection of combinatorics, representation theory, statistical physics, and integrable systems, with potential significant impact on several fields, including applications to shallow water waves, translation in protein synthesis, and scattering amplitudes in supersymmetric Yang-Mills theory. The central structures in the mathematics of the project are Grassmannians, which parameterize subspaces of vector spaces and are ubiquitous in mathematics, appearing variously as projective spaces in projective geometry, compact smooth manifolds in differential geometry, and as a scheme in algebraic geometry. Grassmannians play an important role for spatial recognition in computer vision, in coding and communication theory, in studying shallow water waves in physics, and in the computation of scattering amplitudes of subatomic particles. This project explores features of remarkably rich subsets of real Grassmannians called totally positive and totally non-negative Grassmannians. These structures constitute refinements and extensions of the classical theory of positive definite and positive semi-definite matrices and representation theoretic work in the context of Lie Theory, and their structure will be studied from topological, representation theoretic, and combinatorial points of view. This project also seeks to increase the visibility of women mathematicians via a series of lectures at the University of California, Berkeley given by distinguished women. More concretely, this project concerns several interrelated questions surrounding the positive Grassmannian, Macdonald-Koornwinder polynomials, and the asymmetric exclusion process. In particular, the research will investigate: a new polytopal manifestation of mirror symmetry for flag varieties; the topology of the positive Grassmannian; the structure of soliton solutions to the Kadomtsev?Petviashvili equation coming from the Grassmannian; the combinatorics of the amplituhedron, a new generalization of the positive Grassmannian; and combinatorial formulas for Macdonald-Koornwinder polynomials and asymmetric simple exclusion process probabilities using rhombic tableaux.