Tropical and Algebraic Geometry

Abstract

Over the past few decades, Tropical Geometry has emerged as an enticing subfield of Algebraic Geometry. In Tropical Geometry, one often deals with combinatorial objects which come from algebraic varieties, retaining a lot of their original structure. In this thesis, we look at Tropical Geometry and see how one can use Tropical methods to solve problems in Algebraic Geometry. The primary problem we look at in this thesis is the curve counting problem: how many curves of degree d pass through 3d→1 points in the plane? We study the original proof of this problem, given by Maxim Kontsevich - using the moduli space of stable maps. Then we allude to Mikhailkin’s correspondence result, which showed that these numbers are the same in the Tropical world. We then look at proof of this result in the Tropical world and compare the methods used in this proof to those of Kontsevich’s.