The Algebra, Blueprinted Geometry, and Combinatorics of Matroids

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Lines and planes in a 3-dimensional space are examples of "linear subspaces", a fundamental notion in mathematics. Matroids are discrete analogs of linear subspaces, and they play an increasingly important role in modern mathematics. The PI and Nathan Bowler recently put forth a new point of view in which matroids and linear subspaces are both special cases of a more general class of objects, thus making the analogy between matroids and linear subspaces into something much more precise. Moreover, the PI and Oliver Lorscheid have demonstrated that this more general viewpoint has interesting applications to algebra, algebraic geometry, and combinatorics. This project concerns further development, extensions, and applications of the Baker-Bowler theory. The broader impacts of the proposed work will include supervising postdocs, graduate students, undergraduates, and high school students on research projects, and disseminating mathematics through high-quality written exposition and the organization of conferences and workshops.In more detail, the PI and his collaborators will prove new results about matroid representations, including new results concerning representations of 3-connected matroids and quaternary orientable matroids. They will also extend various aspects of the Baker-Bowler theory to orthogonal matroids (also known as even delta-matroids or Coxeter matroids of type D). Finally, they will develop the rudiments of intersection theory for systems of multivariate polynomials over pastures, extending the PI's work with Lorscheid in the univariate case.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.