The surfaces

Whether we build them, arrange them, knit them, whether it is a matter of making roofs or simply beautiful forms, the manufacturing of surfaces is an extraordinary bridge between mathematics and applications. Let us think of the making of a football. It is often practical questions that pose challenges to theorists, but sometimes, conversely, the abstract beauty of geometry that inspires creators or architects. The properties of surfaces have not finished surprising and raising new mathematical challenges.

For a mathematician, understanding surfaces generally means doing differential geometry or integral calculus. What progress has been made since the ancient Greeks, who had to resort to fearsome tricks adapted to each case encountered! For the practitioner dealing with a specific case, for example the study of the Earth's surface, the question is rather to find specific tools for the particular surface to be studied. From virology to soap bubbles, concrete cases are not lacking. Various questions arise, whose impacts are often unexpected, sometimes leading to considerable applications.

Inventing the nonexistent, exploring unknown worlds, moving "differently"... Surfaces allow theorists to let their imagination run free, to build, for example, objects of a new kind that help understand the trajectory of a billiard ball. Nothing stops creativity, which even goes so far as to explain, with "smooth fractals", how to fit the Earth inside a balloon or a thimble... Mathematics today leads to real miracles!