Toric bundles, tropical bundles, and Khovanskii bases

In this project, recent advances on geometric structures called vector bundles will be leveraged to understand related patterns in several fields of mathematics. Borrowing powerful organizing principles from geometry, the principal investigator aims to establish deep combinatorial patterns and address difficult computational problems involving algebraic equations. This project’s broader impacts include the training of graduate students and opportunities for undergraduates to learn mathematics through rigorous research. The overarching scientific goal of the project is to apply results in the theory of toric vector bundles to investigate several different objects from algebraic geometry and combinatorics. By extending the classical positivity theory of vector bundles to tropical vector bundles, the principal investigator seeks a framework to generalize recent breakthroughs in the study of matroid invariants. The investigator will construct and study moduli spaces for toric vector bundles over various bases, generalizing classical results on the moduli of vector bundles on curves. Finally, the Cox ring of a projectivized toric vector bundle will be studied in a computational manner. The aim of this study is to better understand the construction of the Khovanskii basis algorithm in computational commutative algebra, and the geometric content of the pathological behavior of this algorithm. 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.