From the sixteenth Hilbert problem to tropical geometry

Hilbert’s problem on the topology of algebraic curves and surfaces (the sixteenth problem from the famous list presented at the second International Congress of Mathematicians in 1900) was difficult to formulate. The way it was formulated made it difficult to anticipate that it has been solved. In the first part of the paper the history of the sixteenth Hilbert problem and its solution is presented. The second part of the paper traces one of the ways in which tropical geometry emerged. This article is based on the 4th Takagi Lectures that the author delivered at Kyoto University on June 21, 2008.