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Tropical mathematics. (English) Zbl 1227.14051

Summary: In tropical mathematics, the sum of two numbers is their minimum and the product of two numbers is their usual sum. Many results familiar from algebra and geometry, including the Quadratic Formula, the Fundamental Theorem of Algebra, and Bézout’s Theorem, continue to hold in the tropical world. This article explains how to draw tropical curves, how tropical linear spaces differ from their classical counterparts, and why evolutionary biologists might care. The tropical approach is now an integral part of geometric combinatorics and algebraic geometry. It has also expanded into mathematical physics, number theory, symplectic geometry, computational biology, and beyond. We offer 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 proposed problems are particularly well suited for undergraduate students.

MSC:

14T05 Tropical geometry (MSC2010)
97K20 Combinatorics (educational aspects)
92D15 Problems related to evolution
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