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.