A team of mathematicians has just completed the magnificent ascent to one of the summits of their discipline. On September 11, they published on Arxiv a third preprint (non-peer-reviewed article), out of five, describing the key stages of their climb of the mythical “Langlands program”. The entirety of their work has already been available for several months on the site of one of the first lazarers, Dennis Gaitsgory, research director at the Max Planck Institute for Mathematics in Bonn, Germany. In total, nearly a thousand pages solving a problem analogous to the famous “program” of the Canadian mathematician Robert Langlands who, in 1967, in a letter to his French colleague André Weil, modestly described one of his visions, which a large number of scientists will remain determined, more than fifty years later, to prove. He ended his letter as follows: “If you want to read this as pure speculation I would appreciate it, otherwise I’m sure you have a basket handy.”
This program is also known as “correspondence,” not because it was posted by mail, but because it bridges two “continents” of mathematics. Professor Edward Frenkel of the University of California at Berkeley, who ventured to cross these bridges but is not a member of the team in question, compares it to a “great unification of mathematics”since there is in physics the dream of unifying the visions of quantum mechanics and general relativity, the infinitely small and the infinitely large. Others, to preserve the idea of correspondence, use the metaphor of the Rosetta Stone, whose writings in several languages allowed the deciphering of hieroglyphics. Others prefer to speak of ” dictionary “ : the same idea is expressed in two different languages.
Principle of equivalence
In all cases, the hope, in addition to mapping the mathematical landscape, is that these bridges, Rosetta Stones, dictionaries, etc., will help to resolve questions that remain unanswered. Lost in one continent, you may encounter inhabitants who speak another language and still manage to understand each other. More prosaically, a mathematical problem that is complicated to solve in one region may be “simpler” in another.
As in 1993, when the Englishman Andrew Wiles demonstrated, thanks to this principle of equivalence, a famous theorem, Fermat’s theorem (there are no strictly positive integers). unknown, and AND z as unknownnorth + andnorth = znorthYeah north is greater than or equal to 3). The two continents are, on the one hand, that of number theory and arithmetic, the land of origin of the theorem, and, on the other, that of another theory, the so-called “harmonic” analysis, resulting from the analysis. study of periodic waves, such as sound waves. A classic gymnastics in this “region” is the decomposition of a sound into several “pure” sounds, of different frequencies. The Langlands correspondence generalizes this by establishing an equivalence between harmonic objects, called “automorphic functions” (the sounds), and arithmetic, the frequency of pure sounds.
You have 54.98% of this article left to read. The rest is reserved for subscribers.
