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This will be an introductory talk on Tropical Algebraic Geometry. We will introduce tropical algebra that has two operations, max and plus. The set of extended real numbers with these operations forms a semifield which is called “Tropical Semifield”. We will show how to sketch the tropical lines on the tropical semifield. Then we will introduce a central concept in tropical algebraic geometry, “Amoeba”. This was first mentioned in 1994 by Gelfand, Kapranov and Zelevinsky. This might be seemed as a bridge between classical algebraic geometry and tropical geometry. Sometimes it is helpful to use the tropical counterparts of the algebraic varieties since we can use combinatorics extensively on these simpler objects. We will sketch the tropical line by using amoebas. Then we will pass to a field theoretic notion, Puiseux Series to explain the tropical varieties. We will give the essential definitions and sketch the tropical line by using Puiseux series. Then if time permits, we will try to explain higher dimensional tropical linear varieties. We will also give the relation with the tropical varieties and the analytification of a variety in sense of Berkovich.

Then we will give a point-line incidence structure called k-nets. We will give some open problems related to these Sylvester-Gallai type structures. We will explain how to prove the nonexistence of one case by using tropical algebraic geometry. We will tropicalize the lines and the points and get a contradiction to the existence of the structure.