The Positive Grassmannian: Applications and Generalizations

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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.