It has been said that “You do not really understand something unless you can explain it to your grandmother.” But a mathematician explaining theorems is more than conveying understanding. It is to tell everyone that our childhood dream of being a naturalist and explorer has been granted. Indeed, when a biologist or a mathematician encounters a new being or object, say a cat or a cubic surface, at least two questions arise: What is its structure? What is its place in the big scheme of things? For example, a biologist wants to know if the organism is either an animal or not. Similarly, given a geometric shape, a mathematician wants to know if polynomials can be used to define it or not. Moreover, given an animal, we would like to know its behavior. Is it aggressive? Does it hibernate during the winter? Of course, this division between structure and behavior is ambiguous because they both influence each other. An 800-pound mammal with big claws and huge teeth does not tend to be a tame one, but neither does a four-dimensional manifold. We obtain much information and context from studying both the inner structure of a creature and how it interplays with others.
Walk with me through an example. We can study a domestic cat’s anatomy and say that it has a tapetum lucidum that reflects any light that passes through the retina back into the eye. In another direction, we can say that our cat belongs to the order Carnivora, Kingdom Animalia. Similarly, some of my colleagues focus on describing the characteristics of a cubic surface. They are four-dimensional geometric shapes—two-dimensional if we use complex numbers for moving around—that are cut from space by a single equation of degree three. A classical result dating from 1849 shows that every smooth—that is “good”—cubic surface has 27 straight lines. In another direction, we can try to understand where these surfaces are located about other ones. Recall the basic biological kingdoms: animals, plants, etc. We also have a classification of the complex surfaces started by Max Noether at the end of the nineteenth century and finished by Kodaira and David Mumford in the 1970s. This mathematical result, known as the Enriques-Kodaira classification, organizes all complex algebraic surfaces into ten types such as rational, K3, Enriques, and general type. Among these groups, cubic surfaces belong to the rational ones.
To say that a cat is an animal or a cubic surface is a rational surface is not satisfactory. A finer classification is desired. A biologist will tell us that domestic cats, also known as Felis catus, belong to the genus Felis, which currently has five other species:
Felis chaus (jungle cat)
Felis silvestris (wildcat)
Felis nigripes (black-footed cat)
Felis margarita (sand cat)
Felis bieti (Chinese mountain cat )
All members of the Felis genus share some similarities. For example, their pupils contract to a vertical slit. In our case, a cubic surface belongs to a family of surfaces called del Pezzo. There are nine types of del Pezzo surfaces, and they are labeled by a number—their degree—that varies from one to nine. Cubic surfaces are del Pezzo surfaces of degree three. Del Pezzo surfaces share some characteristics among them, such as a finite number of lines. A del Pezzo surface of degree one has 240 straight lines, while a del Pezzo surface of degree seven has only 3 of them. A deeper result came in the 1960s when Yuri Manin discovered that we can attach a root system to most del Pezzo surfaces. Roots systems are configurations of vectors that play a fundamental role in Lie groups and Lie algebras's theory.
Many subtleties go unmentioned. How do you recognize a cat or a del Pezzo surface? Are there fundamental invariants that help us to distinguish them? If we build a catalog of all cubic surfaces, how big will it be? We are all blessed by the diversity of living organisms on Earth. Fewer people are aware that we are blessed with a richness of mathematical objects and that mathematicians wake up every morning resolved to explore them.