Toric Bundles, Tropical Bundles, and Khovanskii Bases

The proposal addresses open questions in algebraic geometry and combinatorics using new concepts from commutative algebra and tropical geometry. A central role is played by constructions in the theory of buildings, matroid theory, and symbolic computation. Intellectual merit: Algebraic geometry is the study of solution sets of polynomial equations. It has applications in biology, physics, chemistry, and many other sciences, in addition to other fields of mathematics. Toric varieties are a rich class of combinatorically described algebraic varieties, and their connections with polyhedral geometry make them a useful test class for many ideas in algebraic geometry. Vector bundles are a fundamental topic in may of the various forms of geometry and topology. Matroids are a fundamental object in combinatorics, and the recent resolution of the Rota conjecture has made matroids the focus on intense research activity. The theory of buildings initiated in the pioneering works of Tits and Bruhat, is an area of combinatorial representation theory and has deep connections with topology and differential geometry. It aims to unravel hidden combinatorial geometry structures in matrix groups and related spaces. Broader impacts: This project will provide research opportunities for the PI’s graduate students, and con- tribute to the training of the next generation of mathematicians. Parts of this project have already been woven into undergraduate research experiences where they can serve as an introduction to algebraic and convex geometry. The PI directs the UK Math Lab, an undergraduate mathematics research and visualization lab at the University of Kentucky. This project will help the UK Math Lab introduce concepts from geometry and combinatorics to a younger audience, provide mentorship experiences for graduate students, aid undergraduates in carrying out their first research projects, and provide for their travel to conferences to present their ideas to their peers. One of the central objects of this project are Khovanskii bases, which the PI will use to address deep questions in algebraic geometry. These same objects are of importance in various applied contexts, especially physical settings where the goal is the computation of the stationary states of a system of differential equations.