Friday, July 30, 2021

Serious Progress On The Langlands Program And What This Means

Woit's Not Even Wrong blog calls attention to an article in Quanta magazine about an area of research in mathematics called the Langlands program which is probably the most clear explanation of this area of mathematical research that I've ever seen, relating recent dramatic theoretical developments in this area of mathematics that may make it possible to solve several of the leading unsolved mathematical problems in mathematics and physics.

These issues might have been resolved a century and a half earlier, but √Čvariste Galois, the mathematical genius who made the mathematical discoveries that started the ball rolling in this sprawling area of mathematical research died in a duel at the age of twenty, before he could continue developing these ideas. It then took more than a century for other, great, but not quite so epically brilliant, mathematicians to continue to make the next key developments in his work.

Basic Background

One common strategy in advanced mathematics is to take a problem that is too hard to solve, and find a parallel problem in a way that is exactly analogous to the problem you want to solve in the ways that are important but it is easier to solve. You solve the parallel problem, and then you reverse the process so that the answer to the easier to solve parallel problem gives you the answer to the hard to solve original problem.

For example, the Fourier Transform takes certain formulas defined in space and time that is hard to solve or work with, and converts a parallel formula into "frequency space" that is easier to solve and work with. And,  once you solve or simplify the problem in "frequency space" using differential equations, you can then convert that solution or simplification back to the original formula you were studying.

In a more elementary example, most equations can be displayed as a graph of all of the possible solutions of the equation, and sometimes it is easier to work with the graph of the function to readily identify some key properties of the equation, than it is to calculate them as a mathematical formula.

The Langlands Program

The Langlands program is an effort to do a similar kind of transformation to solve problems in number theory and other areas of mathematics. 

The block quotes below, before my quotation of Woit's block, are from the Quanta article with the bold emphasis, and the italic text in brackets, provided by me.

The Langlands program is a sprawling research vision that begins with a simple concern: finding solutions to polynomial equations like x^2 − 2 = 0 and x^4 − 10x^2 + 22 = 0. Solving them means finding the “roots” of the polynomial — the values of x that make the polynomial equal zero (x = ± the square root of 2 for the first example, and x = ± the square root of the sum of 5 and ± the square root of 3, for the second).

By the 1500s mathematicians had discovered tidy formulas for calculating the roots of polynomials whose highest powers are 2, 3 or 4. [Ed. you probably learned the formula for the case of powers of two, called the quadratic equation, in high school.] They then searched for ways to identify the roots of polynomials with variables raised to the power of 5 and beyond. 
But in 1832 the young mathematician √Čvariste Galois discovered the search was fruitless, proving that there are no general methods for calculating the roots of higher-power polynomials.

Galois didn’t stop there, though. In the months before his death in a duel in 1832 at age 20, Galois laid out a new theory of polynomial solutions. Rather than calculating roots exactly — which can’t be done in most cases — he proposed studying the symmetries between roots, which he encoded in a new mathematical object eventually called a Galois group.

In the example x^2 − 2, instead of making the roots explicit, the Galois group emphasizes that the two roots (whatever they are) are mirror images of each other as far as the laws of algebra are concerned.

“Mathematicians had to step away from formulas because usually there were no formulas,” said Brian Conrad of Stanford University. “Computing a Galois group is some measure of computing the relations among the roots.”

Throughout the 20th century mathematicians devised new ways of studying Galois groups. One main strategy involved creating a dictionary translating between the groups and other objects — often functions coming from calculus — and investigating those as a proxy for working with Galois groups directly. This is the basic premise of the Langlands program, which is a broad vision for investigating Galois groups — and really polynomials — through these types of translations.

The Langlands program began in 1967, when its namesake, Robert Langlands, wrote a letter to a famed mathematician named André Weil. Langlands proposed that there should be a way of matching every Galois group with an object called an automorphic form. While Galois groups arise in algebra (reflecting the way you use algebra to solve equations), automorphic forms come from a very different branch of mathematics called analysis, which is an enhanced form of calculus. Mathematical advances from the first half of the 20th century had identified enough similarities between the two to make Langlands suspect a more thorough link. . . .
If mathematicians could prove what came to be called the Langlands correspondence, they could confidently investigate all polynomials using the powerful tools of calculus. The conjectured relationship is so fundamental that its solution may also touch on many of the biggest open problems in number theory, including three of the million-dollar Millennium Prize problems: the Riemann hypothesis, the BSD conjecture and the Hodge conjecture. [Ed. Many other unsolved problems in mathematics have solutions that work if these conjectures can be shown to be accurate.] . . . 
Beginning in the early 1980s Vladimir Drinfeld and later Alexander Beilinson proposed that there should be a way to interpret Langlands’ conjectures in geometric terms. The translation between numbers and geometry is often difficult, but when it works it can crack problems wide open.

To take just one example, a basic question about a number is whether it has a repeated prime factor. The number 12 does: It factors into 2 × 2 × 3, with the 2 occurring twice. The number 15 does not (it factors into 3 × 5).

In general, there’s no quick way of knowing whether a number has a repeated factor. But there is an analogous geometric problem which is much easier.

Polynomials have many of the same properties as numbers: You can add, subtract, multiply and divide them. There’s even a notion of what it means for a polynomial to be “prime.” But unlike numbers, polynomials have a clear geometric guise. You can graph their solutions and study the graphs to gain insights about them.

For instance, if the graph is tangent to the x-axis at any point, you can deduce that the polynomial has a repeated factor (indicated at exactly the point of tangency). It’s just one example of how a murky arithmetic question acquires a visual meaning once converted into its analogue for polynomials.

“You can graph polynomials. You can’t graph a number. And when you graph a [polynomial] it gives you ideas,” said Conrad. “With a number you just have the number.”

The “geometric” Langlands program, as it came to be called, aimed to find geometric objects with properties that could stand in for the Galois groups and automorphic forms in Langlands’ conjectures. Proving an analogous correspondence in this new setting by using geometric tools could give mathematicians more confidence in the original Langlands conjectures and perhaps suggest useful ways of thinking about them.

The New Developments 

A series of papers by Laurent Fargues of the Institute of Mathematics of Jussieu in Paris and Peter Scholze of the University of Bonn, initially developing ideas independently and then collaborating, culminating in a 350 page paper released in February of 2021 which painstakingly resolve a lot of technical issues identified in the last three pages of a previous much shorter 2017 paper that set forth the basic ideas of their collaboration, has now finally filled in a lot of the gaps in the Langlands program.

In 2012, at the age of 24, Scholze invented a new kind of mathematically defined geometric object called a perfectoid space, which he expanded upon in 2014 as he taught an advanced math course for graduate students in which he invented the mathematical content he was teaching as he went along during the semester.
Scholze’s theory was based on special number systems called the p-adics. The “p” in p-adic stands for “prime,” as in prime numbers. For each prime, there is a unique p-adic number system: the 2-adics, the 3-adics, the 5-adics and so on. P-adic numbers have been a central tool in mathematics for over a century. They’re useful as more manageable number systems in which to investigate questions that occur back in the rational numbers (numbers that can be written as a ratio of positive or negative whole numbers), which are unwieldy by comparison.

The virtue of p-adic numbers is that they’re each based on just one single prime. This makes them more straightforward, with more obvious structure, than the rationals, which have an infinitude of primes with no obvious pattern among them. Mathematicians often try to understand basic questions about numbers in the p-adics first, and then take those lessons back to their investigation of the rationals. . . . 

All number systems have a geometric form — the real numbers, for instance, take the form of a line. Scholze’s perfectoid spaces gave a new and more useful geometric form to the p-adic numbers. This enhanced geometry made the p-adics, as seen through his perfectoid spaces, an even more effective way to probe basic number-theoretic phenomena, like questions about the solutions of polynomial equations. . . . 
Fargues together with "Jean-Marc Fontaine in an area of math called p-adic Hodge theory, which focuses on basic arithmetic questions about these numbers" invented "a curve — the Fargues-Fontaine curve — whose points each represented a version of an important object called a p-adic ring."

Fargues and Scholze were both in Berkley for a semester long session of the Mathematical Science Research Institute where they learned about each others work and started to collaborate with each other. They generalized the concept of the Fargues-Fontaine curve so it could be used to describe the kind of p-adic number system problems that Scholze was using his perfectoid spaces to deal with, and called these generalized structures "diamonds" which represent p-adic groups.

"Diamonds" make it possible to link Galois groups and specific mathematical structures that are easier to work with, and the collaborates proved with these new structures that there is always a specific Galois group that can be associated with a specific p-adic group. This proves half of a special case of the Langlands correspondence known as the "local Langlands correspondence". The other half of the local Langlands correspondence would show to transform a specific Galois group into a p-adic group, and progress in proving this half of the local Langlands correspondence seems likely. What is a "diamond"?
Imagine that you start with an unorganized collection of points — a “cloud of dust,” in Scholze’s words — that you want to glue together in just the right way to assemble the object you’re looking for. The theory Fargues and Scholze developed provides exact mathematical directions for performing that gluing and certifies that, in the end, you will get the Fargues-Fontaine curve. And this time, it’s defined in just the right way for the task at hand — addressing the local Langlands correspondence.
Progress in proving the existence of this large class of special cases of the local Langlands correspondence, in turn, makes prospects of generalizing the local Langlands correspondence to the global Langlands correspondence which applies to all rational numbers, rather than just p-adic groups.

The local and global Langlands correspondence relate to their corresponding local and global geometric Langlands correspondence with geometric objects known as "sheaves", the theory of which was brought to its current level by Alexander Grothendieck in the 1950s. 

The part of the local Langlands correspondence which was been proved by Fargues and Scholze has also been extended to the geometric Langlands correspondence by finding a way to translate their findings made with "diamonds" into descriptions using "sheaves."

So basically, Fargues and Scholze over the last decade, have taken the mush of speculation and stray thoughts and concepts that have been swirling around creating lots of heat but little light over the last half century around the promise of the Langlands program, and have solved the first of four main components of the ultimate goal. They have also provided a much more focused roadmap to solving the next three pieces of this problem.

Implications

Recall, as noted above in bold, that the prize at the end of the day for solving all four pieces is a solution to a large share of the most important unsolved problems in mathematics, many of which have practical applications in physics as well.

Woit's commentary on it is as follows;

Quanta magazine has a good article about the dramatic Fargues-Scholze result linking geometry and number theory....

[These are] extremely interesting topics indicating a deep unity of number theory, geometry and physics. They’re also not topics easy to say much about in a blog posting. In the Fargues-Scholze case that’s partly because the new ideas they have come up with relating arithmetic and geometry are ones I don’t understand very well at all (although I hope to learn more about them in the future). The connections they have found between representation theory, arithmetic geometry, and geometric Langlands are very new and it will likely be quite a few years before they are well understood and their implications well-developed. . . .

There is a fairly short path now potentially connecting fundamental unifying ideas in number theory and geometry to our best fundamental theories in physics (and seminars on arithmetic geometry and QFT are now a thing). 
The Fargues-Scholze work relates arithmetic and the central objects in geometric Langlands involving categories of bundles over curves. These categories in turn are related (in work of Witten and collaborators) to 4d TQFTs based on twistings of N=4 super Yang-Mills. This sort of 4d QFT involves much the same ingredients as 4d QFTs describing the Standard Model and gravity. For some better indication of the relation of number theory to this sort of QFT, a good source is David Ben-Zvi’s lectures this past semester (see here and here). 
I’m hopeful that the ideas about twistors and QFT in Euclidean signature discussed here will provide a close connection of such 4d QFTs to the Standard Model and gravity (more to come on this topic in the near future).

Of course, until the Langlands correspondence can be proven, this is all just a status report on a large scale, long term mathematical research effort that doesn't have much to show for it. 

But, this research is a potential sleeper wildcard that could lead to a rapid rush of major theoretical discoveries in mathematics and physics if it can be worked out, which is something that could easily happen in the next several years to a decade.

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