14 1 A Taste of Things to Come
Fig. 1.1. A cubic curve in the tropical projective plane. Mikhalkin ,
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 .
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.