The Biggest Project in Modern Mathematics

The Biggest Project in Modern Mathematics

Imagine a map of the mathematical world. It's the product of thousands of years of human ingenuity, from the Babylonians to Riemann to the present day. In this world is everything we know about numbers, shapes and their relationships. There are many mathematical continents, each with its own language and culture. One with a distinguished history is number theory. It's an unpredictable land of lush forests full of opportunity with little secrets hidden everywhere. Here, they speak the language of arithmetic. Another continent is called harmonic analysis. It's a place of smooth curves, symmetry and repeating patterns. Here, mathematical objects speak the language of signals and waves. For most of history these two continents have remained distant strangers. Mathematicians didn't think there would be much to gain from trying to travel between them. But in the last half century something striking has happened: We've discovered glimpses of an enormous bridge that seems to connect the continents. As mathematicians build segments of this bridge, they can pass seemingly intractable problems across it, illuminating them in the process. The full bridge is called the Langlands Program. It's one of the biggest projects in modern mathematical research. Some even like to call it a grand unified theory of mathematics. I'm Alex Kontorovich, and I'll be your guide as we explore some of the patterns and symmetries suggested by the Langlands Program, ideas that imply deep powerful and beautiful connections between the most fundamental continents of the mathematical world. in 1967, a 30-year-old-mathematician named Robert Langlands wrote a deceptively modest letter to his colleague, the famous French number theorist Andre Weil. "If you are willing to read this as pure speculation, I would appreciate that," the letter said. "If not, I'm sure you have a wastebasket handy." To Weil's surprise, the letter contained a series of striking conjectures that predicted something very unlikely: A correspondence between two objects from completely different fields of math. What Langlands suggested seemed stranger than telepathy. How could two objects indigenous to different mathematical continents evolved to behave in exactly the same way? This is the question at the heart of the Langlands Program. To understand this mysterious connection we first have to meet two mathematicians who studied these objects from opposite shores. This is Srinivasa Ramanujan. In 1916, the self-taught Indian math prodigy got very interested in a particular function, now known as the Ramanujan discriminant function. Put simply, this function multiplies infinitely many terms together. In Ramanujan's time, this was already known to be a special type of function called a modular form. To see modular forms it's not enough to look at the real numbers; we need to treat the input as a complex number which will produce a complex output. When we do this, the function starts to reveal its mesmerizing symmetries. Modular forms are some of the most bizarre objects in mathematics. They satisfy so many internal symmetries that it's surprising that they exist at all. Ramanujan wasn't the first to study the symmetries of modular forms, but he was the first to imagine whether they had any connection to the distant shores of number theory. He did this by studying the function from a different angle. What would happen, Ramanujan wanted to know, if you multiplied all of its terms out and collected the coefficients? When he did this, Ramanujan noticed that the coefficients have a special kind of predictive power: If you know all of the function's prime coefficients, then you can use them to figure out the rest. For example, if you find this coefficient, which is negative 24, and this coefficient, which happens to be 252, then I can immediately tell you what the coefficient of x to the sixth will be, because two times three is six. This pattern always seemed to hold, but Ramanujan couldn't prove why. He became obsessed with these numbers, and continued to experiment with them, eventually making more remarkable conjectures on their behavior. But they were so difficult to prove that they sat collecting dust. Nearly six decades later, Belgian mathematician Pierre Delign came up with a brilliant proof of Ramanujan's conjecture, earning Delign a Fields Medal. Delign used the key insight from Langland's conjectures, called functoriality, to bypass the bridge from harmonic analysis to number theory. But the bridge goes in both directions. Now, it's time to travel over to the shores of number theory, where the rest of our story will take shape. In 1637, the French lawyer turned hobbyist mathematician Pierre de Fermat scribbled an equation in his copy of Diophantus's Arithmetica. He never offered a proof, claiming that it was too marvelous for the narrow margins to contain. The theorem featured a polynomial equation, a fundamental object native to the continent of number theory. These are some of the most basic equations you can write down. Just variables, with exponents, that are positive whole numbers. You may remember the Pythagorean theorem from school, that a right triangle with sides a, b and hypotenuse c will satisfy this equation. There are lots of integer solutions. Here's the note that Fermat wrote in the margins of Arithmetica: Unlike the pythagorean theorem, these polynomial equations will never have natural number solutions, except of course if one of the variables is zero. This tantalizingly simple claim came to be known as Fermat's Last Theorem. For 350 years, generations of mathematicians struggled to prove or disprove this conjecture. Then, in the 1990s Princeton mathematician Andrew Wiles shocked the world with a breakthrough. Standing on the shores of number theory, he began dreaming of a bridge to harmonic analysis. To understand how wiles built this bridge, we'll first need to meet a special type of polynomial equation called an elliptic curve. Here is an example of an elliptic curve. You'll notice that it only has two variables, x and y. Let's take a look at our elliptic curve graphed on the cartesian plane. This shows all the pairs of real numbers, x and y, that satisfy our equation. But we're not just interested in the real solutions; we also want to make the problem more interesting by restricting the domain to find solutions in the rational numbers (that is, fractions) and in the integers. To study these solutions, it's useful to consider our elliptic curve from another perspective. We can use a fantastic tool from number theory called modular arithmetic. It's a way of counting integers using only the remainders. We use this kind of math every day when we tell time on a 12 hour clock. If someone says, "Let's meet at 1500," you know they really mean 3 pm, because both 15 and 3, when divided by 12, leave the same remainder. Now, what if we decided to change our 12 hour clock to a different number, say 31? When we do this we change what's called the modulus. So when the modulus is 31, what's 6 squared? Well, it should be 36, but on our clock we see the number 5 because we're 5 hours past high noon. Here's the funny thing about modular arithmetic: Equations like this, which don't have rational solutions, may have modular solutions, since 6 squared is the same as 5 modular 31 which means that x equals 6 solves the equation. Now let's go back to our elliptic curve and try to find modular solutions. What we really care about is how many solutions there are. Here is the graph of all of the modular solutions of our equation. The dots are the pairs x and y for which x cubed plus x plus 17 when divided by 31 leaves the same remainder as y squared. For a modulus n, let's call the number of solutions b sub n. For example, when n equals 31, we have 24 solutions. We can compute these values b sub n for lots of different moduli n, that is, clocks with different hours. This gives us an infinite sequence. Now we can make a function by taking this infinite sequence, multiplying each term by a power of x, and summing them all up. The result is called an infinite power series, which is like a polynomial but with an infinite number of terms. Right about now you might be wondering, "What's the point of making this function?" When Wiles began building the bridge to harmonic analysis, he used some important scaffolding built decades earlier by three mathematicians named Taniyama, Shimura and Weil. They had predicted that for any elliptic curve, the resulting function would be a modular form, an object with the same kinds of symmetries that Ramanujan studied In the land of harmonic analysis. So since we just created a function from an elliptic curve, shouldn't it be a modular form in disguise? Remember that Ramanujan started with a function that he knew was a modular form and tried to learn something about its coefficients. Here, we have a function and we know its coefficients, since they come from counting solutions on the elliptic curve with different clocks. But what we don't know is whether our function is, in reality, a modular form. So let's graph it on the unit disk and see what happens. Incredibly, it does seem to be a modular form. But is this just a fluke? This is the question Wiles had to answer to finish building the bridge. He had to prove that every elliptic curve is intimately related to a modular form. But wait a second, why are we talking about elliptic curves when we started out trying to understand Fermat's last theorem, which is a completely different type of polynomial equation? German mathematician Gerhard Frey made a key connection here. He observed that if Fermat's equation did have a hypothetical solution, in other words, a counter example, that is a to the p plus b to the p was indeed equal to c to the p for p greater than two, then you could make this elliptic curve, which seems to have extremely bizarre properties. In particular, when you create the infinite power series out of it, it would not have the symmetries required to be a modular form. This means that if Fermat's last theorem were false, then the Taniyama-Shimura-Weil conjecture would also be false, because the counter-example to Fermat's last theorem would make an elliptic curve which is not modular. So when Wiles and his student Richard Taylor proved that every elliptic curve does indeed produce an infinite power series which is modular, they also showed that Frey's elliptic curve can't exist -- which means that a solution to fermat's equation can't exist either! And that's how Wiles proved Fermat's last theorem. It's one of the craziest proofs by contradiction ever. By showing that elliptic curves give rise to modular forms, Wiles built the bridge from number theory to harmonic analysis. And by studying the special predictive power of the coefficients in modular forms, Ramanujan (and later Delign) used the bridge from harmonic analysis to number theory. These proofs are some of the most monumental achievements in modern mathematics, but they're also just a tiny sliver of the vast construction project that is the Langlands Program. Langland's ideas have traveled to the shores of algebraic geometry, representation theory and quantum physics, where mathematicians will be building bridges for years to come. Nobody knows how far Langland's vision extends, but many mathematicians agree that it has the potential to solve some of the most intractable problems of our time. Someday the Langland's Program may reveal the deepest symmetries between many different continents, a kind of grand unified theory of the mathematical world that answers the most fundamental questions we have about numbers.