Project “TROPGEO”, 2010-2014, Area of activity: Geometry (Mathematics)
The goal of this project is to develop Tropical Geometry, a newly emerging kind of algebraic geometry. It is expected to be more powerful than Classical Geometry in a range of applications (particularly in Physics-minded applications). In the same time it is significantly simpler in several mathematical aspects. In the last decade a number of initial applications of this new geometry has appeared with a success, particularly in the framework of the so-called Gromov-Witten theory, based on curves, i.e. 1-dimensional algebraic varieties. The new subject became known as Tropical Geometry since algebraically it is a based on the so-called “Tropical Calculus” of Computer Science. In the tropical world the curves are metric graphs, sometimes enhanced with additional structure. Stepping forward from our recent successes in set-up and application of Tropical Geometry we plan to continue this work. Particularly we plan to advance the following challenging lines of research: Solve several classical complex enumerative problems, particularly compute Zeuthen's characteristic numbers. Develop tropical homology theories. Advance the theory of amoebas and coamoebas (algae) of algebraic varieties. Advance understanding of real algebraic geometry. Establish direct relation between Feynman diagrams and tropical curves. Break the Gromov-Witten barrier in Enumerative Geometry. Develop birational tropical geometry in higher dimensions. These directions are intrinsically related in their scope and possible methodology. Some of the proposed goals are very ambitious, but even partial advances would mean a big step forward.