Tropical Geometry

Tropical geometry is an emerging area of algebraic geometry in which a variety is studied via its combinatorial shadow, known as the tropical variety. At its most basic, tropical geometry is geometry over the tropical semiring, where multiplication is replaced by addition and addition is replaced by minimum. Tropical polynomials are thus piecewise linear functions: 3x^2+2y^2 becomes min(2x+3,2y+2).A (complex affine) algebraic variety is the set of common solutions in the complex numbers to a set of polynomial equations. Tropical geometry turns a variety into a polyhedral fan, called the tropical variety, which is a combinatorial object.The overarching aim of this project is to determine which invariants of a variety can be computed from its tropical variety. In particular, the first goal of the project is to determine when the nef and effective cones of a variety can be determined via tropical methods. These cones are an important invariant of the variety coming from birational geometry. The second goal of the project is to understand when the Chow or cohomology rings of the variety can be determined from the variety. The final goal is to apply this understanding to the tropical space of stable maps, by realizing these spaces as tropicalizations of the original spaces.

The major beneficiaries of this project include the tropical geometry community, birational geometers, and the UK tropical mathematics community. Each of these groups will benefit through the research results and examples developed in the course of the work. In addition the PDRA and PhD will gain transferable skills (conference organisation and computer skills). To ensure this impact, the research results of the project will be disseminated widely, with papers circulated electronically (on Maclagan's webpage and the Arxiv) prior to publication, and code submitted as Macaulay 2 packages. In addition, all three members of the group (Maclagan, the PDRA, and the PhD student) will give conference and seminar presentations on the results, both locally in the UK and internationally. Finally, a workshop to be held at Warwick at the end of the second year of the project will also help show-case these results.