Geometry(ies) and groups
When Évariste Galois died at the age of 20 after a duel in 1832, no one would have predict how deep was the revolution implied by his work. The reason was that no one could understand the mathematics of this romantic figure. Hopefully, history regenerated them.
From equations to groups
One learns at school the formulas that gives the solutions (if they exist) of a polynomial equation of degree 2:
- \(ax^2+bx+ c=0\).
Whereas similar (but complicate) formulas exists in degree 3 and 4, nothing general is known for polynomial equations of degree greater than 5:
- \(x^n+a_{n-1} x^{n-1} + \dots +a_0=0\) with \(n\geq 5\).
The great achievement of Galois’s work has been to prove that no such formula exists. This breakthrough relies on a hidden structure of the equation that he called the «groupe de l’équation». These groups describe somehow the symmetries of the solutions.
The work of Galois was only published in 1846 and understood and accepted by the mathematical community years after. However, at the end of the 19th century, the notion of group had been generalized and had been effectively used in any area of mathematics. In particular, Galois theory has been used to solve some 2100 years old problems such as squaring the circle, angle trisection or doubling the cube.
The definition of a group is rather abstract but we can say that a group is a set of symmetries of a mathematical object such as an equation, a curve, a surface, a space of functions etc.
Since Galois, geometry took a great benefit of researches in group theory. However, little by little, it appeared also that geometry should be use to make progress in abstract group theory. This is remarkable because geometry (even the most advanced geometry) is a very concrete way to do mathematics: using pictures in your mind or on the black board. Mathematically, the idea of using concrete mathematics (geometry) to study abstract objects (groups) is beautiful.
Today, geometric group theory is a flourishing field in which groups and geometries are studied in parallel. The community of geometric group theorist is growing fast and hundreds of papers are published every year under this label.
From groups to geometries
The 19th century is the century during which the antic foundations of mathematics are questioned. In particular, the mathematical works of Gauss, Bolyai, Lobachevsky and Riemann, promulgated the idea that Euclidean geometry is only one geometry among others. Then, under the influence of Lie and Klein it became usual to study the geometry of a space through its groups of symmetries.
The following pattern is recurrent in the history of mathematical progress: 1) a new concept is discovered through a concrete problem, 2) the new concept is taken out of its context and studied abstractly, 3) the concrete applications of the abstract concept provide both new discoveries and problems. And so on so forth. Of course, the notion of concreteness is relative. Usually a mathematician considers that concrete mathematics start around his own field of expertise…
This is what happened with groups during the 20th century: they became a subject of research by themselves, new applications were discovered in computer science and in cryptography as these new sciences arose.