Tropical geometry is a polyhedral shadow of algebraic geometry and naturally lies in the intersection of geometric combinatorics and algebraic geometry. It has a wide range of applications in enumerative geometry, mirror symmetry, computational algebra, optimization, algebraic statistics, and computational biology. The strengths of tropical methods come from the fact that the tropical objects are intrinsically combinatorial, and computations can go farther on combinatorial objects than on algebro-geometric objects. Better understanding of combinatorial structures in tropical geometry has led to new algorithms and formulas in enumerative geometry and computational algebra. Moreover, tropical geometric objects have rich combinatorial structures that also arise naturally in discrete geometry and combinatorial algebra, such as graphs, subdivisions and triangulations of polytopes, fiber polytopes, matroid theory, space of phylogenetic trees, and cellular resolutions of monomial ideals, just to name a few. The project aims to understand combinatorial and topological structures and further the development of new algorithms in three main directions: tropical varieties, tropical curves, and tropical semialgebraic sets.
Tropicalizations of algebraic varieties are piecewise linear, so tropicalization replaces difficult algebraic computations with easier polyhedral computations. Tropical methods can be used to develop algorithms and software for solving classical problems in computational commutative algebra. Although tropical varieties have been around for years, very few families of them have known homology. This project includes a study of the combinatorial topology of natural families of tropical varieties such as tropicalizations of complete intersections, determinantal varieties, Grassmannians, and resultants. A difficulty is the lack of examples to check conjectures and develop intuitions on. The PI proposes to build a library of examples for the aforementioned families of varieties and complete classifications when feasible. Tropical curves are metric graphs that naturally arise in graph theory and electrical network theory. They are simple combinatorial objects, yet they are powerful enough for proving new theorems about classical algebraic curves. The proposal aims at a better understanding of projective embeddings and ramifications of tropical curves. Application of tropical geometry to semialgebraic sets and optimization is a promising but under-explored direction. This project aims to develop new algorithms for tropical convexity and understand the combinatorics of tropical semialgebraic sets. Many parts of this project are suitable for involvement of students and for interdisciplinary collaborations. Research tools and software will be developed for discrete geometry and computational algebra. The computational methods may be useful in other areas such as semialgebraic optimization, algebraic statistics, and computational biology.