This is what Robert Langlands did, but his ambition went deeper than simply joining a few islands. Instead, the Langlands Program that he initiated in the late 1960s has become an attempt to find the mechanism by which we could build bridges between many islands, no matter how unrelated they may seem.
Robert Langlands at his office in Princeton, 1999. Photo by Jeff Mozzochi.
Langlands is now emeritus professor of mathematics at the Institute for Advanced Study in Princeton, where he occupies the office formerly held by Albert Einstein. A man of amazing talent and vision, he was born in 1936 and grew up in a small town near Vancouver; his parents had a millwork business. One of the striking things about Langlands is his fluency in many languages: English, French, German, Russian, and Turkish, even though he didn’t speak any languages besides his native English before he entered college.1
I have had the opportunity to collaborate with Langlands closely in recent years, and we have often corresponded in Russian. At some point he sent me the list of Russian authors that he had read in the original. The list was so extensive that it seemed he may well have read more of my native Russian literature than I have. I often wonder whether Langlands’ unusual language abilities have something to do with his power to bring together different mathematical cultures.
The key point of the Langlands Program is the concept of symmetry that is already familiar to us. We have talked about symmetry in geometry: for example, any rotation is a symmetry of a round table. Our study of these symmetries has led us to the notion of a group. We then saw that groups appear in mathematics in different guises: as groups of rotations, braid groups, and so on. We have also seen that groups were instrumental in classifying elementary particles and predicting the existence of quarks. The groups that are relevant to the Langlands Program appear in the study of numbers.
To explain this, we need to talk first about the numbers that we encounter in our everyday life. Each of us was born in a particular year, lives in a house that has a particular number on the street, has a phone number, a PIN to access a bank account at the ATM, and so forth. All of these numbers have something in common: each of them is obtained by adding number 1 to itself a certain number of times: 1 + 1 is 2, 1 + 1 + 1 is 3, and so on. These are called the natural numbers.
We also have the number 0, and the negative numbers: −1, −2, −3,... As we discussed in Chapter 5, these numbers go by the name “integers.” So an integer is a natural number, or number 0, or the negative of a natural number.
We also encounter slightly more general numbers. A price, in dollars and cents, is often represented like this: $2.59, meaning two dollars and fifty-nine cents. This is the same as 2 plus the fraction 59/100, or 59 times 1/100. Here 1/100 means the quantity that being added to itself 100 times gives us 1. Numbers of this kind are called rational numbers, or fractions.
A good example of a rational number is a quarter; mathematically, it is represented by the fraction 1/4. More generally, for any two integers m and n we can form the fraction m/n. If m and n have a common divisor, say d (that is to say, m = dm′ and n = dn′), then we can cancel out d and write m′/n′ instead of m/n. For example, 1/4 can also be represented as 25/100, and that’s why Americans can say that a quarter is the same thing as 25 cents.
The vast majority of the numbers we encounter in our everyday life situations are these fractions, or rational numbers. But there are also numbers that are not rational. An example is the square root of 2, which we write as follows: . It is the number whose square is equal to 2. Geometrically, is the length of the hypotenuse of the right triangle with legs of length 1.
It turns out we cannot represent it as m/n, where m and n are two natural numbers.2 However, we can approximate it by rational numbers if we write the first few digits of its decimal form: 1.4142, then 1.41421, then 1.414213, and so on. But no matter how many decimal digits we retain, this will be an approximation – there will be more digits to follow. No finite decimal number will ever do justice to .
Since is the length of the hypotenuse of the above triangle, we know that this number is out there. But it just does not fit the numerical system of rational numbers.
There are many other numbers like that, such as or the cubic root of 2. We need to develop a systematic way to add these numbers to the rational numbers. Think of the rational numbers as a cup of tea. We can drink it by itself, but our experience will be enhanced if we mix in sugar, milk, honey, various spices – and these are like the numbers , , etc.
Let’s try to mix in . This will be the equivalent of adding a cube of sugar to our cup of tea. So we drop in the rationals and see what kind of numerical system we obtain. Surely, we want to be able to multiply the numbers within this new numerical system, so we have to include all numbers that are products of rational numbers and . These have the form . So our numerical system must include all fractions (these are the rational numbers) and all numbers of the form . But we also want to be able to add them to each other, so we also have to include the sums
The collection of all numbers of this form is already “self-contained,” in the sense that we can perform all the usual operations on them – addition, subtraction, multiplication, and division – and the result will also be a number of the same form.3 This is our cup of tea with the cube of sugar fully mixed with the tea.
It turns out that this new numerical system has a hidden property that the rational numbers didn’t have. This property will be our portal into the magical world of numbers. Namely, it turns out that this numerical system has symmetries.
By a “symmetry” I mean here a rule that assigns a new number to whatever number we begin with. In other words, a given symmetry transforms each number to another number from the same numerical system. We will say that a symmetry is a rule by which each number “goes” to some other number. This rule should be compatible with the operations of addition, subtraction, multiplication, and division. It is not clear yet why we should care about the symmetries of a numerical system. Please bear with me and you will see why momentarily.
Our numerical system has the identity symmetry, the rule by which every number goes to itself. This is like the rotation of a table by 0 degrees, under which every point of the table goes to itself.
It turns out that our numerical system also has a non-trivial symmetry. To explain what it is, let’s observe that is a solution of the equation x2 = 2. Indeed, if we substitute for x, we obtain an equality. But this equation actually has two solutions: one of them is and the other is . And we have in fact added both of them to the rational numbers when we constructed our new numerical system. Switching these two solutions, we obtain a symmetry of this numerical system.*
To illustrate this more fully in terms of our tea cup analogy, let’s modify it slightly. Let’s say that we drop a cube of white sugar and a cube of brown sugar in our cup and mix them with the tea. The former is like and the latter is like . Clearly, exchanging them will not change the resulting cup of tea. Likewise, exchanging and will be a symmetry of our numerical system.
Under this exchange, rational numbers remain unchanged.4 Therefore, the number of the form will go to the number . In other words, in every number we simply change the sign in front of and leave everything else the same.5
You see, our new numerical system is like a butterfly: the numbers are like the scales of a butterfly, and the symmetry of these numbers exchanging and is like the symmetry of the butterfly exchanging its wings.
More generally, we can consider other equations in the variable x instead of x2 = 2; for example, the cubic equation x3 − x + 1 = 0. If the solutions of such an equation are not rational numbers (as is the case for the above equations), then we can adjoin them to the rational numbers. We can also adjoin to the rational numbers the solutions of several such equations at once. This way we obtain many different numerical systems, or, as mathematicians call them, number fields. The word “field” refers to the fact that this numerical system is closed under the operations o
f addition, subtraction, multiplication, and division.
Just like the number field obtained by adjoining , general number fields possess symmetries compatible with these operations. The symmetries of a given number field can be applied one after another (composed with each other), just like symmetries of a geometric object. It is not surprising then that these symmetries form a group. This group is called the Galois group of the number field,6 in honor of the French mathematician Évariste Galois.
The story of Galois is one of the most romantic and fascinating stories about mathematicians ever told. A child prodigy, he made groundbreaking discoveries very young. And then he died in a duel at the age of twenty. There are different views on what was the reason for the duel, which happened on May 31, 1832: some say there was a woman involved, and some say it was because of his political activities. Certainly, Galois was uncompromising in expressing his political views, and he managed to upset many people during his short life.
It was literally on the eve of his death that, writing frantically in a candlelit room in the middle of the night, he completed his manuscript outlining his ideas about symmetries of numbers. It was in essence his love letter to humanity in which he shared with us the dazzling discoveries he had made. Indeed, the symmetry groups Galois discovered, which now carry his name, are the wonders of our world, like the Egyptian pyramids or the Hanging Gardens of Babylon. The difference is that we don’t have to travel to another continent or through time to find them. They are right at our fingertips, wherever we are. And it’s not just their beauty that is captivating; so is their high potency for real-world applications.
Alas, Galois was far ahead of his time. His ideas were so radical that his contemporaries could not understand them at first. His papers were twice rejected by the French Academy of Sciences, and it took almost fifty years for his work to be published and appreciated by other mathematicians. Nevertheless, it is now considered as one of the pillars of modern mathematics.
What Galois had done was bring the idea of symmetry, intuitively familiar to us in geometry, to the forefront of number theory. What’s more, he showed symmetry’s amazing power.
Before Galois, mathematicians focused on trying to discover explicit formulas for solutions of equations like x2 = 2 and x3 − x + 1 = 0, called polynomial equations. Sadly, this is how we are still taught at school, even though two centuries have passed since Galois’ death. For example, we are required to memorize a formula for solutions of a general quadratic equation (that is, of degree 2)
in terms of its coefficients a, b, c. I won’t write this formula here so as not to trigger any unpleasant memories. All we need to know about it now is that it involves taking the square root.
Likewise, there is a similar, but more complicated, formula for a general cubic equation (of degree 3)
in terms of its coefficients a, b, c, d, which involves cubic roots. The task of solving a polynomial equation in terms of radicals (that is, square roots, cubic roots, and so forth) is quickly becoming more and more complicated as the degree of the equation grows.
The general formula for the solutions of the quadratic equations was already known to the Persian mathematician Al-Khwarizmi in the ninth century (the word “algebra” originated from the word “al-jabr,” which appears in the title of his book). Formulas for solutions of the cubic and quartic (degree 4) equations were discovered in the first half of the sixteenth century. Naturally, the next target was a quintic equation (of degree 5). Prior to Galois, many mathematicians had been desperately trying to find a formula for its solutions for almost 300 years, to no avail. But Galois realized that they had been asking the wrong question. Instead, he said, we should focus on the group of symmetries of the number field obtained by adjoining the solutions of this equation to the rational numbers – this is what we now call the Galois group.
The question of describing the Galois group turns out to be much more tractable than the question of writing an explicit formula for the solutions. One can say something meaningful about this group even without knowing what the solutions are. And from this one can then infer important information about the solutions. In fact, Galois was able to show that a formula for solutions in terms of radicals (that is, square roots, cubic roots, and so on) exists if and only if the corresponding Galois group has a particularly simple structure: is what mathematicians now call a solvable group. For quadratic, cubic, and quartic equations, the Galois groups are always solvable. That’s why solutions of these equations may be written in terms of the radicals. But Galois showed that the group of symmetries of a typical quintic equation (or an equation of a higher degree) is not solvable. This immediately implies that there is no formula for solutions of these equations in terms of radicals.7
I won’t get into the details of this proof, but let’s consider a couple of examples of Galois groups to give you an idea what these groups look like. We have already described the Galois group in the case of the equation x2 = 2. This equation has two solutions, and , which we adjoin to the rational numbers. The Galois group of the resulting number field8 then consists of two elements: the identity and the symmetry exchanging and .
As our next example, consider a cubic equation written above, and suppose that its coefficients are rational numbers, but all of its three solutions are irrational. We then construct a new number field by adjoining these solutions to the rational numbers. It’s like adding three different ingredients to our cup of tea: say, a cube of sugar, a dash of milk, and a spoonful of honey. Under any symmetry of this number field (the cup of tea with these ingredients added), the cubic equation won’t change because its coefficients are rational numbers, which are preserved by symmetries. Hence each solution of the cubic equation (one of the three ingredients) will necessarily go to another solution. This observation allows us to describe the Galois group of symmetries of this number field in terms of permutations of these three solutions. The main point is that we obtain this description without writing down any formulas for the solutions.9
Similarly, the Galois group of symmetries of the number field obtained by adjoining all solutions of an arbitrary polynomial equation to the rational numbers may also be described in terms of permutations of these solutions (there will be n solutions for a polynomial equation of degree n whose solutions are all distinct and not rational). This way we can infer a lot of information about the equation without expressing its solutions in terms of the coefficients.10
Galois’ work is a great example of the power of a mathematical insight. Galois did not solve the problem of finding a formula for solutions of polynomial equations in the sense in which it was understood. He hacked the problem! He reformulated it, bent and warped it, looked at it in a totally different light. And his brilliant insight has forever changed the way people think about numbers and equations.
And then, 150 years later, Langlands took these ideas much farther. In 1967, he came up with revolutionary insights tying together the theory of Galois groups and another area of mathematics called harmonic analysis. These two areas, which seem light years apart, turned out to be closely related. Langlands, then in his early thirties, summarized his ideas in a letter to the eminent mathematician André Weil. Copies were widely circulated among mathematicians at the time.11 The letter’s cover note is remarkable for its understatement:12
Professor Weil: In response to your invitation to come and talk, I wrote the enclosed letter. After I wrote it I realized there was hardly a statement in it of which I was certain. If you are willing to read it as pure speculation I would appreciate that; if not – I am sure you have a waste basket handy.
What followed was the beginning of a groundbreaking theory that forever changed the way we think about mathematics. Thus, the Langlands Program was born.
Several generations of mathematicians have dedicated their lives to solving the problems put forward by Langlands. What was it that so inspired them? The answer is coming up in the next chapter.
*Note that here and below I
use a minus sign (a dash) to represent negative numbers, rather than a hyphen. This conforms to the standard mathematical notation. In fact, there isn’t really any difference between the two because -N = 0 − N.
Chapter 8
Magic Numbers
When we first talked about symmetries in Chapter 2, we saw that representations of a group named SU(3) govern the behavior of elementary particles. The focus of the Langlands Program is also on representations of a group, but this time it is the Galois group of symmetries of a number field of the kind discussed in the previous chapter. It turns out that these representations form the “source code” of a number field, carrying all essential information about numbers.
Langlands’ marvelous idea was that we can extract this information from objects of an entirely different nature: the so-called automorphic functions, which come from another field of mathematics called harmonic analysis. The roots of harmonic analysis are in the study of harmonics, which are the basic sound waves whose frequencies are multiples of each other. The idea is that a general sound wave is a superposition of harmonics, the way the sound of a symphony is a superposition of the harmonics corresponding to the notes played by various instruments. Mathematically, this means expressing a given function as a superposition of the functions describing harmonics, such as the familiar trigonometric functions sine and cosine. Automorphic functions are more sophisticated versions of these familiar harmonics. There are powerful analytic methods for doing calculations with these automorphic functions. And Langlands’ surprising insight was that we can use these functions to learn about much more difficult questions in number theory. This way we find a hidden harmony of numbers.
Love and Math Page 9