Tropical Mathematics

These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the “tropical approach” in mathematics, which has gotten a lot of attention recently in combinatorics, algebraic geometry and related fields. It offers an an elementary introduction to this subject, touching upon Arithmetic, Polynomials, Curves, Phylogenetics and Linear Spaces. Each section ends with a suggestion for further research. The bibliography contains numerous references for further reading in this field. The adjective “tropical” was coined by French mathematicians, including Jean-Eric Pin [19], in the honor of their Brazilian colleague Imre Simon [23], who was one of the pioneers in min-plus algebra. There is no deeper meaning in the adjective “tropical”. It simply stands for the French view of Brazil.