"Fear will make the local systems submissive." So spoke Grand Moff Tarkin, the commander of the Death Star, in the 38th minute of the first "Star Wars" film. What does this sentence, in its original English, have to do with a mathematical article? Is it a nerdy "Abandon all hope" to all those who know nothing about category theory and feel at home in topological spaces?
Indeed, without such knowledge, it is impossible to read what Dennis Gaitsgory presented last year together with Sam Raskin of Yale University. It is nevertheless spectacular.
The two mathematicians thus concluded a five-part series with a total of more than 900 printed pages proving a theorem called the "Geometric Langlands Conjecture."
Gaitsgory, Director of the Max Planck Institute for Mathematics in Bonn since 2021, had already worked on the problem as a student in Tel Aviv. In the end, he led a nine-person team, together with his former doctoral student Raskin, they finally succeeded in proving the hypothesis. For this, Gaitsgory was awarded the Breakthrough Prize for Mathematics, endowed with a whopping three million dollars, last night in Los Angeles – and Raskin was awarded one of the six "New Horizons" prizes for young mathematicians. The Breakthrough Prizes – there are also ones for life sciences and physics – are sponsored by tech billionaires, and their presentation is modeled after the Oscars. The idea is to create publicity. But is that even possible in the case of the geometric Langlands conjecture?
But perhaps, with some scientific advances, it is ultimately inevitable. Anyone who takes note of news from genetic research or astrophysics today can do so because, even without the ability to read specialist literature, they have an inkling of the molecular basis of life or the possibility of a curvature of space, concepts that were no less alien to early modern people than the geometric Langlands conjecture is to most science enthusiasts today.
The award-winning theorem is part of a broader project, the so-called Langlands Program, one aspect of which already attracted public attention.
That was in 1995, when the British scientist Andrew Wiles proved Fermat's conjecture, which states that the equation x + y = z for integers x, y, and z to the power of integer n can only be fulfilled if the exponent n is greater than 2.
What Wiles actually proved back then was a conjecture, the correctness of which implies Fermat's conjecture. It was first proposed in 1957 by the Japanese Yutaka Taniyama and Goro Shimura, as well as the Frenchman André Weil – and it leads to something astonishing: If you consider a specific equation and insert integers with certain properties, you can determine the number of cases in which the equation holds true using a function that is familiar to so-called Fourier analysis.
Thus, the conjecture of Taniyama, Shimura, and Weil asserted the existence of a connection between two completely different fields of mathematics: on the one hand, number theory, which has been concerned with integers and their building blocks, prime numbers, since antiquity; and on the other, the realm of continuous functions and their decomposition into sine and cosine waves of different frequencies and amplitudes, which the French mathematician Joseph Fourier described in 1822.
Is there something more general behind this? This was the question Robert Langlands, a Canadian native who was then working at Princeton, asked himself in 1967, wondering whether difficult problems in number theory could be translated into less difficult problems in Fourier analysis.
He outlined a search for such relationships in a letter to André Weil. He wrote: "I would be happy if you would like to read this as pure speculation; if not, I'm sure you have a wastebasket handy."
"That was the understatement of the century," Berkeley mathematician Edward Frenkel once commented on this note. Indeed, the Langlands program developed into a veritable mathematical quest for the Holy Grail in the 1970s. "The entire number theory group in Tel Aviv was working on the original Langlands program," Dennis Gaitsgory recalled in an interview with the Frankfurter Allgemeine Zeitung about his first year as a doctoral student. "But I only heard about the geometric Langlands conjecture when Alexander Beilinson, who now teaches in Chicago, gave a lecture to us. I was overwhelmed. It was so beautiful."
André Weil had already had an inkling of a connection to geometry in 1940. He had seen analogies between three different materials. Thematic disciplines, including number theory and geometry, have noticed this. The latter, however, refers less to the theory of points, lines, and surfaces in ordinary space that can be described with real coordinates, but rather to so-called Riemann surfaces, which are based on an extended concept of numbers, that of complex numbers. Beilinson and his Chicago colleague Vladimir Drinfeld, among others, began searching in the 1980s for a correspondence between Riemann surfaces and a corresponding analogue to Fourier analysis. Gaitsgory and his team have now shown, in a very general way, that such a correspondence exists and what exactly it consists of.
Just as number theory is related to Fourier analysis in certain cases, such as the Taniyama-Shimura-Weil theorem, the geometry of Riemann surfaces is also related to something in which objects can be decomposed into their components. "But that has nothing to do with Fourier anymore," explains Gaitsgory. "It's merely an analogy to this in algebra." The role of sine and cosine waves is thus assumed here by special examples from a class of objects that mathematicians call "sheaves." Only with the proof that the sheaves in question also possess the necessary properties to fulfill this task was the harvest, so to speak, reaped.
But with the now presented proof, Robert Langland's vision is by no means exhausted, not even for geometry. "What we considered was the so-called global unramified case," explains Gaitsgory. "There is also the ramified case. Then you have to manage to combine the two, and only then is the geometric Langlands theory complete."
In addition, so-called local Langlands relationships between small disks around points on Riemann surfaces and objects from number theory, the p-adic numbers [A], are being researched. Gaitsgory's institute colleague Peter Scholze, together with a French colleague, made important progress here in 2021. Scholze was supposed to receive a "New Horizons" Breakthrough Prize in 2016, but declined. Two years later, he received the Fields Medal, the highest award in mathematics, for his theory of a geometry over p-adic numbers. Furthermore, there are efforts to develop Langlands correspondences leading to quantum field theories in physics.
When the question arises about the practical use of his proof, however, Gaitsgory shakes his head. "That's pure mathematics," he says. If one wants to speak of utility, then it consists at most in methods developed along the way. These might be useful elsewhere, he says, but hardly directly in solving one of the great puzzles. "I don't think the proof of the geometric Langlands conjecture helps in any way with the Riemann hypothesis." A "grand unified theory of mathematics" is by no means the goal of the Langlands program either. "I don't know where that comes from," Gaitsgory comments on this phrase, which is popular in media reports. After all, the point isn't to find a mathematics that is somehow more fundamental than, say, geometry or number theory, but rather to establish connections between them, or rather to suitable analogues to Fourier analysis. Gaitsgory prefers the image of bridges being built between different continents of mathematics, connecting one to the other and back again. These bridge-building projects are awe-inspiring enough.
But they aren't meant to inspire fear, only to invite wonder at their potential. So why was Grand Moff Tarkin involved? "When we were working on the paper, we had named several chapters after various 'Star Wars' episodes," Gaitsgory says. "Just kidding, because I was watching 'Star Wars' with my younger son at the time." The Tarkin quote, however, does have a meaning: The central focus of this final paper, he says, was the proof of a specific equality statement. "We proved this with a trick, and it involved the behavior of a specific mathematical object, the so-called space of local systems. It was made compliant, that is, restricted so that this equality no longer had any choice. It had to be true." ULF VON RAUCHHAUPT” [B]
A. P-adic numbers are a system of numbers, each associated with a prime number, that extend the rational numbers. They are like the real numbers but with a different notion of "closeness" determined by the chosen prime number. Essentially, p-adic numbers are written as infinite series of digits in base p, extending to the left instead of the right like decimals.
1. Extension of Rational Numbers:
p-adic numbers build upon the rational numbers (fractions).
For each prime number 'p', there's a distinct field of p-adic numbers, denoted as Qp.
2. Different Notion of Closeness:
In p-adic numbers, two numbers are considered close if their digits agree for a long way to the left (in base p).
This contrasts with the real numbers, where closeness is based on how close the numbers are on a number line.
For example, in 5-adic numbers, 25 is considered closer to 0 than to 10 because 25 = 5 * 5, and 5 is a "small" number in this system.
3. p-adic Representation:
p-adic numbers are written as infinite series of the form: a₀ + a₁p + a₂p² + a₃p³ + ... , where aᵢ are integers from 0 to p-1.
The "digits" are to the left of the "decimal point," which is defined by the powers of p.
4. Applications:
P-adic numbers have applications in number theory, algebra, and analysis.
They can be used to solve problems that are difficult or impossible to solve with real numbers.
Examples include Hensel's lemma for finding roots of polynomials and Mahler's theorem.
5. Key Differences from Real Numbers:
P-adic numbers are not ordered like real numbers.
They are a non-Archimedean field, meaning that they don't have the same properties as the real numbers.
In simpler terms:
Imagine you're looking at numbers in a different way. Instead of focusing on how close numbers are to each other on a number line (like we do with real numbers), you focus on how many digits they share in common when written in base 'p'. This leads to a different kind of "closeness" and a different way of understanding the world of numbers.
B. Das Understatement des Jahrhunderts: Der Mathematiker Dennis Gaitsgory hat zwei weit entfernte Felder seines Fachs verbunden. Dafür erhält er jetzt einen der höchstdotierten Wissenschaftspreise. Frankfurter Allgemeine Zeitung; Frankfurt. 07 Apr 2025: 12.
Komentarų nėra:
Rašyti komentarą