Tropical Singularities

01.03.2016 bis 01.12.2017

In tropical geometry, algebraic varieties are degenerated to polyhedral complexes called tropical varieties. In spite of the degeneration, many properties of an algebraic variety can be read off its tropical counterpart, and many theorems continue to hold on the tropical side. Tropical geometry thus provides an approach to study questions in algebraic geometry by means of discrete mathematics and combinatorics. Tropical geometry has been succesfully applied to computational problems and problems in enumerative geometry, in particular real enumerative questions.

To really make good use of the new tool, many questions of translating nature have to be answered. The central question for this project is: what is the correct translation of the concept of a singularity to the tropical world?
This question is quite natural to ask and has consequently interested several authors recently. The fact that it is hard to answer in general makes it even more intriguing. The aim of this project is to considerably improve the understanding of tropical singularities, to apply the results to questions in enumerative geometry, and to investigate the connections of this subject to questions in combinatorics.