14 1 A Taste of Things to Come Fig. 1.1. A cubic curve in the tropical projective plane. Mikhalkin [385], reproduced with permission. We also have a full-blown tropical algebraic geometry, where curves in the plane are made from pieces of straight lines (Fig- ure 1.1)—quite like the graph of the absolute value function y = |x|, the starting point of our discussion. Tropical mathematics is amusing, but is it relevant? Yes, and very much so. Moreover, it currently is experiencing an explo- sive growth. There are intrinsic mathematical reasons for tropical mathematics to exist, but its present flourishing is largely moti- vated by applications. One application is mathematical genomics: tropical geometry captures the essential properties of “distance” between species in the phylogenic tree. Another is theoretical physics: tropical mathematics can be treated as a result of the so-called Maslov dequantization of tradi- tional mathematics over numerical fields as the Planck constant tends to zero taking imaginary values [380]. A third application is computer science and the theory of time- dependent systems, like queuing networks (where tropical mathe- matics is known under the name of a (max, +)-algebra. The ratio- nale behind this class of applications is an observation so simple and banal that it has a certain ironic flavor. Indeed, we do not nor- mally multiply time by time instead, we either add two intervals of time (which corresponds to consecutive execution of two pro- cesses) or compare the lengths of two intervals—to decide which process ends earlier. Therefore tropical mathematics is mathemat- ics of time—which also explains its applications to genomics: phy- logenic trees grow in time, and the geometry of phylogenic trees reflects the geometry of time.

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