The Biggest Project in Modern Mathematics

Source: The Biggest Project in Modern Mathematics, Quanta Magazine, 13:18, uploaded 2022-06-01, category Mathematics, playlist index 1491.

Alex Kontorovich introduces the Langlands Program through two areas of mathematics that seem to speak different languages. Number theory studies arithmetic and the hidden structure of integers. Harmonic analysis studies waves, repeating patterns and symmetry. The program proposes that objects from these areas can correspond in precise ways, so that a difficult problem in one area becomes legible after it crosses into the other.

Two mathematical worlds and a conjectural connection

The video opens with a visual map of mathematics, built over thousands of years from Babylonian mathematics through Riemann and into current research. Number theory appears as the older home of arithmetic. Harmonic analysis works with smooth curves, signals and waves. Their subjects seem far apart, and mathematicians spent much of history with little reason to expect a useful route between them.

In 1967, the 30-year-old Robert Langlands wrote to the French number theorist André Weil. Langlands described his ideas as speculation and added that Weil probably had a wastebasket ready if he disliked them. The letter proposed a correspondence between mathematical objects from different fields. The video treats this correspondence as the question at the centre of the Langlands Program: why should objects that arise from unrelated constructions show the same behaviour?

The correspondence is a conjecture rather than one finished theorem. The Langlands Program gathers a large family of related predictions about symmetries and links between parts of mathematics. The video uses the image of a bridge because results can travel in both directions. A theorem about a function from harmonic analysis can reveal information about numbers, whilst a number-theoretic construction can produce an object with the symmetries of harmonic analysis.

Ramanujan and the predictive coefficients of a modular form

The first route begins with Srinivasa Ramanujan. In 1916, the self-taught Indian mathematician studied a function now called the Ramanujan discriminant function. It multiplies infinitely many terms together and belongs to a class called modular forms.

To see what makes a modular form unusual, the video moves from real inputs to complex inputs and outputs. The function then displays a dense pattern of internal symmetries. Ramanujan studied the function through its coefficients after multiplying out the terms. He found that the coefficients attached to prime numbers seemed to predict the others. If two coefficients are known, including the coefficient -24 and the coefficient 252, their relation can determine the coefficient of $x^6$ because $2 \times 3 = 6$. The exact rule extends this pattern across the coefficients.

Ramanujan could observe the pattern without proving why it held. He kept experimenting and produced further conjectures about the numbers. The problems then remained open for decades. Nearly sixty years later, the Belgian mathematician Pierre Deligne proved Ramanujan’s conjecture and received a Fields Medal. The video presents Deligne’s proof as an early success for the Langlands viewpoint. Deligne used functoriality, one of the ideas in Langlands’ conjectures, to transfer the problem between harmonic analysis and number theory.

That movement supplies one direction of the connection. The video then turns to a famous number-theory problem to show the bridge working in the other direction.

Fermat’s equation and Wiles’ route through elliptic curves

In 1637, Pierre de Fermat wrote a claim in the margin of his copy of Diophantus’s Arithmetica. The claim concerned equations of the form

For the Pythagorean equation, where $n=2$, many positive whole-number solutions exist. Fermat claimed that for exponents greater than two, the equation has no positive whole-number solutions. He wrote that his proof was too large for the margin. The claim became Fermat’s Last Theorem and remained unproved for 350 years.

In the 1990s, Andrew Wiles approached the theorem from number theory and looked for a route into harmonic analysis. The route begins with an elliptic curve, a special kind of polynomial equation with two variables. The video first plots the real solutions as points on the Cartesian plane, then restricts attention to rational and integer solutions. The curve becomes useful because number theory can study its solutions through several related systems of arithmetic.

One of those systems is modular arithmetic. A 12-hour clock treats 15 and 3 as equivalent because both leave the same remainder after division by 12. If the modulus becomes 31, then $6^2 = 36$ becomes 5 because 36 leaves remainder 5. An equation with no rational solution can still have a solution modulo a particular number. The video uses the point $x=6$ modulo 31 as the example.

For the elliptic curve, mathematicians count the solutions for many moduli. Calling the number of solutions $b_n$ produces an infinite sequence. The sequence can then become the coefficients of an infinite power series, a polynomial with infinitely many terms. This construction gives the curve a new mathematical object whose coefficients come from counting its modular solutions.

The Taniyama–Shimura–Weil conjecture predicted that the power series from every elliptic curve would be a modular form. That would make the new object share the symmetries found in Ramanujan’s work. The problem therefore changes shape. The coefficients are known from the curve, while the unknown question is whether the resulting function has the symmetries of a modular form. The video plots an example on the unit disk and shows that it appears to fit. Wiles had to prove that this resemblance held for every elliptic curve.

Frey’s curve and the proof by contradiction

Gerhard Frey supplied the link back to Fermat’s equation. Suppose a counterexample to Fermat’s Last Theorem existed, so that positive integers $a$, $b$ and $c$ satisfied $a^p + b^p = c^p$ for a prime exponent $p$ greater than two. Frey observed that such a solution could produce an elliptic curve with an unusual property.

The power series constructed from Frey’s curve would fail to have the symmetries required of a modular form. A counterexample to Fermat would therefore produce an elliptic curve that was not modular. The Taniyama–Shimura–Weil conjecture said that every elliptic curve was modular. The two statements could not both hold.

Wiles and his student Richard Taylor proved the needed modularity result for elliptic curves. Their result ruled out Frey’s curve, which ruled out the hypothetical counterexample to Fermat’s equation. Fermat’s Last Theorem followed through this chain of implications. The proof never attacks the original equation by searching through its possible solutions. It proves a broader statement about elliptic curves and lets that statement eliminate the object that a counterexample would have created.

The video closes by returning to the two directions of the Langlands connection. Ramanujan and Deligne used modular forms to learn about number-theoretic coefficients. Wiles and Taylor used elliptic curves to establish a modular form connection and settle Fermat’s theorem. Those results occupy a small part of the larger program, which the video says has reached into algebraic geometry, representation theory and quantum physics. Its possible range remains unknown. Some mathematicians describe the hoped-for network of connections as a grand unified theory of mathematics.

Limits of the explainer

This is a visual introduction, not a proof of the Langlands conjectures, Deligne’s theorem, the Taniyama–Shimura–Weil conjecture or Wiles’ modularity theorem. The transcript gives the main logical chain and the examples needed to follow it, while leaving the technical arguments out. Claims about the Langlands Program eventually solving some of mathematics’ hardest problems remain forecasts attributed to the video and to the mathematicians it cites. The map-and-bridge presentation explains the relation between the fields, although the underlying correspondences involve precise definitions that the 13-minute format does not provide.

Further reading / references

• What Is the Langlands Program?, Quanta Magazine article linked in the video’s description.