An ambitious programme to unify disparate areas of mathematics was set out some fifty years ago by Robert Langlands of the Institute for Advanced Study in Princeton. The “Langlands Program” (LP) is a set of deep conjectures that attempt to build bridges between certain algebraic and analytical objects.
Canadian mathematician Robert Langlands, who formulated a series of far-reaching conjectures [image from Wikimedia Commons].
Substantial strides have been made in the ensuing decades. For example, in 2009 the Vietnamese mathematician Ngo Bao Chau proved a result known as the Fundamental Lemma. He was awarded a Fields Medal the following year for this work. However, most of the conjectures of the Langlands Program remain unproven.
The LP has been called the Grand Unified Theory of mathematics, as it holds the possibility of linking several crucial areas of maths. It is also intimately related to modern physics, which seeks to reconcile quantum field theory and general relativity.
Number Theory
The arithmetic that we learn at school is the beginning of the sub-field of mathematics called number theory. Although one of the most ancient branches, it is still vibrant and intensively studied today.
Carl Friedrich Gauss gave the subject of number theory a major impetus when he published his Disquisitiones Arithmeticae in 1798. He was just 21 years old at this time. To Gauss is attributed the quotation “Mathematics is the queen of the sciences and number theory is the queen of mathematics.”
The central concern of number theory is the system of whole numbers or integers. One focus of number theory is the behaviour of the prime numbers. The Prime Number Theorem was proved using methods from analysis, a branch of mathematics not obviously related to number theory. This led on to the emergence of analytic number theory
Another focus of number theory is Diophantine equations, polynomial equations with integer coefficients, where we are interested in integer solutions. The earliest link between Diophantine equations and geometry was the search for Pythagorean triplets, integer solutions of
x2 + y2 = z2
corresponding to right-angled triangles with sides of integral length.
The most famous Diophantine equation is Fermat’s equation
xn + yn = zn
For n = 1 this is trivial: once integers x and y are given, we let z be their sum. When n = 2, the equation has many solutions in whole numbers, Pythagorean triplets such as (3, 4, 5). For n > 2, there are no integer solutions. This is Fermat’s Last Theorem (FLT), proved in the 1990’s by Andrew Wiles.
By applying the methods of algebraic geometry to the solution of Diophantine equations a new subject, Diophantine geometry, has emerged. Wiles’ proof of the FLT is a spectacular application of this merger of geometry and number theory. It falls within the ambit of the Langlands Program.
“Love and Math”, a recent book about the Langlands Program, and its author Edward Frenkel.
Elliptic Curves and Solutions over Finite Fields
Quadratic equations like the Pythagorean equation, involving only squares but no higher powers, have either an infinite number of rational solutions or none at all. Elliptic curves like y2 = x ( x2 – 1), with terms of cubic order, can have either a finite or an infinite number of rational solutions.
It is surprising that elliptic curves have a group structure associated with them (Silverman and Tate, 1992). Given two points a and b on an elliptic curve, we can find a third one such that a + b + c = 0. This gives us an algebraic framework for analysing the properties of such continuous curves.
We may seek solutions of Diophantine equations in number systems other than the integers. As long as addition, subtraction and multiplication make sense, polynomials can be defined. Thus, we may restrict our solutions to a finite field such as Fp, the integers modulo p,
{ 0, 1, 2, … , p – 1 }
where p is a prime number. Taking the example in Frenkel’s book (see Sources below),
y2 + y = x3 – x2,
it is easy to show that there are four solutions in the field F5, namely
(x, y) in { (0, 0), (0, 4), (1, 0), (1, 4) }
Since we can consider a polynomial equation over a finite field or over the integers, real or complex numbers, the equation provides a bridge between these domains: it links number theory and analysis.
The number of integer solutions of an elliptic curve in Fp can be encapsulated in a generating function, a function of a complex variable. These generating functions have periodic properties. They are, in a sense, generalizations of the sine and cosine functions. This provides an unexpected link between harmonic analysis and number theory.
Thus, a polynomial equation can connect numbers and surfaces, or number theory and analysis. Curves over finite fields are the objects that link number theory and Riemann surfaces. Such connections are at the heart of the Langlands Program.
André Weil and Robert Langlands
André Weil was one of the giants of twentieth century mathematics. He was a founding member of the group of French mathematicians who published under the sobriquet Bourbaki. While in prison in 1940, as a result of refusing military service, Weil formulated some conjectures about deep connections between number theory and geometry. He explained his ideas in clear and simple language in a letter to his sister, the noted philosopher Simone Weil.
André Weil (1906-1998)
Weil found that the number of solutions for each field (each p) could be encapsulated in a single function now called Weil’s zeta-function. He formulated a number of conjectures about this function. One of these was proved in 1964 by Alexander Grothendieck, the others by Pierre Deligne some ten years later.
Robert Langlands wrote to Weil, suggesting much more sweeping interconnections. In this letter, he modestly wrote that, if Weil was not interested in the contents, “I am sure that you have a waste basket handy”. The ideas set out in the letter led to the Langlands Program.
Langlands conjectured that many hard problems in number theory (such as counting the number of solutions of polynomials modulo primes) could be solved by means of harmonic analysis. This implied strong connections between areas of mathematics which were thought to be unrelated. And some of these have been found.
Langlands made a connection between representations of the Galois groups of number fields – studied in number theory – and automorphic functions – studied in harmonic analysis. This was an unexpected and surprising relationship between two fields of mathematics thought to be far removed from each other.
The LP also has some surprising links to quantum physics. The Langlands dual group is linked to gauge theories (e.g. electrodynamics). In a recent paper, Kapustin and Witten (2007) wrote: “We aim to show how this program (a geometric version of LP) can be understood as a chapter in quantum field theory”.
A Rosetta Stone
Patterns emerging in disparate areas can indicate a mysterious underlying structure of all mathematics. The Langlands Program is like a Rosetta Stone of mathematics. In his recent book, Edward Frenkel expressed this view on the LP: “I believe that it holds the keys to understanding what mathematics is really about, far beyond the original Langlands conjectures.”
Sources
Edward Frenkel, 2013: Love and Math: The Heard of Hidden Reality. Basic Books. New York, 292pp. ISBN: 978-0-465-05074-1.
Anton Kapustin and Edward Witten, 2007: Electric-Magnetic Duality And The Geometric Langlands Program. Comms. in Num. Th. and Phys. Vol 1, No 1, 1–236.
Joseph H Silverman and John Tate, 1992: Rational Points on Elliptic Curves. Springer, ISBN: 978-1-4419-3101-6.
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