The Positive Grassmannian: Applications and Generalizations

该项目探讨了位于组合学，表示论，统计物理和可积系统交叉点的结构，对几个领域具有潜在的重大影响，包括浅水波，蛋白质合成中的翻译和超对称杨米尔斯理论中的散射振幅。 该项目数学中的中心结构是格拉斯曼（Grassmannian），它是向量空间的参数化子空间，在数学中无处不在，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。Grassmannian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振幅的计算中发挥着重要作用。这个项目探讨了非常丰富的真实的格拉斯曼子集的特征，称为完全积极和完全非消极的格拉斯曼。这些结构构成了正定和半正定矩阵的经典理论的改进和扩展，以及李论背景下的表示理论工作，它们的结构将从拓扑，表示理论和组合的角度进行研究。该项目还力求通过由杰出妇女在加州大学伯克利分校举办的一系列讲座，提高女数学家的知名度。更具体地说，这个项目涉及几个相互关联的问题，围绕积极的格拉斯曼，麦克唐纳-库恩温德多项式，和不对称排斥过程。特别是，研究将调查：一个新的多面体表现为镜像对称的旗品种;拓扑结构的积极格拉斯曼孤子解决方案的Kadomtsev？Petviashvili方程来自格拉斯曼;振幅面体的组合学，一个新的推广的正格拉斯曼;和组合公式的麦克唐纳-Koornwinder多项式和不对称简单排除过程概率使用菱形tableaux。

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.