Tropical geometry for Nagata's conjecture and Legendrian curves

In this thesis, tropical methods in singularity theory and legendrian geometry are developed; tropical modifications are surveyed and several technical statements about them are proven. In more details, if a planar algebraic curve over a valuation field contains an $m$-fold point, then there is a certain collection of faces in the subdivision of the Newton polygon of this curve, with total area of order $m^2$. This estimate can be applied in Nagata's type questions for curves. Then, the notion of a tropical point of multiplicity $m$ is revisited. With some additional assumptions, the tropicalization of a complex legendrian curve in $mathbb CP^3$ is proven to enjoy a certain divisibility property. Finally, with help of tropical modifications, the tropical Weil reciprocity law is proven and several restrictions on the realizability of non-transversal intersection of tropical varieties are obtained