Combinatorial and Computational Algebraic Geometry

My project lies in the area of Commutative Algebra and its interactions with Algebraic Geometry, Tropical Geometry, Combinatorics, and Convex Geometry. The main goal is to associate convex polytopes to algebraic varieties such that significant geometric properties of the variety can be read off from their polytopes. A toric variety is a certain algebraic variety modeled on a convex polytope. My main goal is to develop new and unifying tools to extend the tools from toric varieties to general varieties via toric degenerations. I will first develop combinatorial tools to construct points in tropical varieties and study their associated Gröbner degenerations. Then, I find explicit characterization for such points leading to toric degenerations, that is their corresponding polynomial ideal is binomial and prime. Furthermore, I will study the relations among toric degenerations of a given variety by studying their associated polytopes. In particular, I will study the combinatorial mutations of these polytopes. My goal is to determine isomorphic degenerations and characterize them. Finally, I will provide algorithms to compute tropicalization and toric degenerations for specific families of varieties. As a particular case, I plan to study the Grassmannians and flag varieties.