Artists, science-fiction enthusiasts, but also engineers, physicists and mathematicians, the fourth dimension is an intriguing and fascinating concept. Today's mathematicians juggle easily with 4-dimensional spaces, which arise naturally in many situations. In the introduction to his famous 1895 dissertation on Analysis situs, Poincaré wrote: “Geometry with n dimensions has a real object; no one doubts that today. The beings of hyperspace are susceptible of precise definitions like those of ordinary space, and if we cannot represent them to ourselves, we can conceive and study them." Far from being a mind game, it wasn't long after Poincaré and the work of 19th-century mathematicians that Einstein developed his breathtakingflante theory of gravitation, a geometric theory of space-time.

There are few mathematical concepts that percolate beyond the scientific sphere. The butterfly effect is a counter-example, and we come across it in literature, films and commercials... The general idea is that a small disturbance can have far-reaching consequences. But chaos theory, mainly developed in the second half of the twentieth century, goes far beyond this, offering theoretical and practical tools when others rely on the simple will of chance. Mathematicians of the 21st century have made precise conjectures about the typical behavior of a dynamic system: from the periodic movements of the Ancients, we have gradually arrived at the idea that, in general, there must be coexistence between chaos and statistical stability, an absolutely remarkable phenomenon.

Symmetries are all around us: butterfly wings, crystals and the mosaics of Granada's Alhambra. The mathematical world also abounds in symmetries, some of them obvious, such as those of the square or the sphere, but also far more mysterious and fascinating. It wasn't until the 19th century that the concept of the group emerged, a concept that has become ubiquitous throughout mathematics, formalizing the idea of symmetries. If you know a bit about group theory, you'll have a much better understanding, for example, of the motions involved in solving a Rubik's cube, the symmetries of algebraic equations, geometry and so on. For reasons that are still largely mysterious, there seem to be extremely profound links between certain symmetries of an arithmetical nature and harmonic analysis, a field of mathematics that grew out of Fourier's work on the heat equation.

Between 1895 and 1904, Henri Poincaré founded algebraic topology (then called Analysis situs) by publishing a series of six groundbreaking memoirs. These seminal texts are written in Poincaré's inimitable style: ideas abound... and errors abound... Together, they represent just over 300 pages of exceptional mathematics and, 120 years on, the content of these memoirs is not only still relevant today, but a highly recommended passage for any apprentice topologist. Today, topology is a research field in its own right, and is involved in all aspects of mathematics. Its applications go far beyond theoretical mathematics, for example in topological data analysis, a methodological approach at the heart of “big data”.

The homotopic theory of types, at the crossroads of mathematics, theoretical computer science and logic, is a new research theme that has been in full bloom in recent years. The subject is truly intriguing, all the more so as it questions the very foundations of mathematics, and is directly linked to another subject that has repercussions even in industry: the certification of proofs and programs. To certifier a program is, in a way, to provide proof of proof via a step-by-step mechanical vérification, and this is precisely the raison d'être of software like Coq, which is handled almost like a video game in which the player must demonstrate goals using the hypotheses at his disposal and an arsenal of tactics. It's mathematically fascinating and somewhat confusing, but the thrills are guaranteed!

Some branches of mathematics lend themselves well to drawings, and we've known, at least since Descartes, that “geometry is the art of right reasoning on wrong figures”. So often, a drawing is better than a long speech to get an idea across, even if some authors make it a point of honor not to include any illustrations at all! The advent of computers has enabled enthusiastic amateurs to take up the challenge too, and today we don't hesitate to speak of fractal art, to cite just one emblematic example. If illustrations are often an excellent means of communication and a very good teaching aid, digital images can also be a tool for mathematicians in their own research work, particularly when they suggest that one mathematical fact rather than another seems to be true: they then provide a clue to help demonstrate the correct theorem!