Brief introduction to tropical geometry

The paper consists of lecture notes for a mini-course given by the authors at the Gökova Geometry & Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.

The goal of these lectures is to give a basic introduction to tropical geometry focusing on some of its particularly simple and visual aspects. The first section is devoted to tropical arithmetic and its relations to classical arithmetic. The second section reviews tropical curves in R2. The content of these two sections is quite standard, and we refer to [Bru09, Bru12, BS14a] for their extended versions. Section 3 contains a tropical version of the combinatorial patchworking construction for plane curves, as well as a tropical reformulation of Haas’ theorem. Section 4 presents some enumerative applications of tropical geometry, as well as the floor diagram technique. Section 5 looks at general tropical subvarieties of Rn and their approximation by complex algebraic varieties. Section 6 is devoted to a basic study of tropical curves inside non-singular affine tropical surfaces. Finally, in Section 7 we define abstract tropical manifolds and review their homology theories.