By Paul Basken

The mathematical concepts behind the "fundamental lemma," as postulated by Robert P. Langlands and proven by Ngo Bao Chau, are exceedingly difficult to explain. So much so that when asked, many top mathematicians declined even to make an attempt. And among those who agreed to try, the discussions quickly bogged down in dense terminology.

That said, there are some basic concepts that can be explained, at least in general terms that will mean something to most readers.

First of all, math at all levels can reveal an amazing number of deep interconnections and profound patterns, many of which aren't obvious at first glance, even to experienced mathematicians. One very simple example from elementary geometry is the theorem of the standard 30-60-90 right triangle, in which the line opposite the triangle's 90-degree angle is known to be always exactly twice as long as the line opposite its 30-degree angle.

Upon a bit of reflection, that theorem does turn out to be fairly easy to prove, merely by placing two such triangles against each other so that they form a single 60-60-60 triangle with all three sides the same length. That makes it clear, visually, that the longest side of the original triangle is twice the length of its shortest side.

That simple triangle is a classic example of the surprising harmony and profound beauty that keep popping up throughout all levels of mathematics, from the simplest geometric figures to the most complex multilevel equations.

And that complexity can often arise from what may seem like some of the briefest and simplest mathematical statements. One of the most famous examples, which comes from a main division of mathematics known as number theory, is Fermat's Last Theorem. It was proposed by Pierre de Fermat in 1637 and states that no three positive integers A, B, and C can satisfy the equation A to the power of N plus B to the power of N equals C to the power of N, for any integer N that is greater than two. That theorem wasn't proved until 1995, by Andrew J. Wiles, a professor of mathematics at Princeton University whose attack on the problem was aided by insights into the deep cohesive structure of mathematics suggested by Mr. Langlands.

Number theory generally refers to the mathematics of quantities that can be represented by whole numbers. That branch is certainly broad, as the three-century exploration of Fermat's Last Theorem suggests. And yet number theory is also considered largely separate from another branch of mathematics called analysis, or calculus, in which mathematicians make exact calculations of complex systems by considering quantities in infinitely tiny increments. Common functions in calculus include the relatively simple and commonly understood trigonometric functions of sine and cosine, which are used to define the relationship between the angles and lengths of a triangle. Sine and cosine functions are also the basic functions describing wave-shaped patterns.

It also helps, when venturing into the world of Mr. Langlands and Mr. Ngo, to appreciate a few other key mathematical concepts. Two of those concepts are "symmetry" and "groups." As with much of math, symmetry is a simple idea with potentially complex implications and applications. It basically refers to a change in a system—such as a movement in time or space, or perhaps some other dimension—that does not alter its properties. In the previous example of a 60-60-60 triangle, also known as an equilateral triangle, the figure can be rotated by 120 degrees any number of times without changing its properties. Each of those rotations is considered a symmetry of the triangle. In other words, symmetry is a pretty common mathematical condition.

A group is, simply, the collection of all symmetries of a given object. Our equilateral triangle has three rotational symmetries, or ways it can be turned while maintaining its same shape and position. More complex mathematical objects and systems have more symmetries, a fairly simple notion that can grow very complex when considering complicated functions and multiple real and theoretical dimensions.

And, finally, there's a specific kind of group analysis known as harmonic analysis, which refers to the study of wave formations. A wave, too, is a simple and familiar concept with complex implications. Harmonic analysis is, essentially, the study of groups of waves with variations in their length, height, frequency, and phase. Multiple waves are often compared by their relative phase, which refers to the degree to which they travel in or out of sync with one another. Harmonic analysis is especially important in the study of light and other forms of electromagnetic energy, such as radio signals, which are best represented by waves.

Mr. Langlands, now an emeritus professor at the Institute for Advanced Study, in New Jersey, proposed the concept that came to be known as the fundamental lemma in 1979. It is an element of an overall structure known as the Langlands Program, a "grand unifying theory" that shows common links between number theory and harmonic analysis, said Edward Frenkel, a professor of mathematics at the University of California at Berkeley.

In other words, the Langlands Program refers to a surprising set of ways in which mathematical behaviors found in number theory appear to also exist in harmonic analysis. And the fundamental lemma was seen by Mr. Langlands as just one piece of that overall structure, or a subset of those surprising patterns, that occurs within the world of harmonic analysis.

The solution developed by Mr. Ngo, therefore, is a series of equations and explanations running nearly 200 pages that proves the mathematical relationships within harmonic analysis necessary for sustaining the Langlands Program. The fundamental lemma essentially states that certain integrals, called orbital integrals, associated with two different groups and integration domains, or orbits, inside those groups, are equal to each other, Mr. Frenkel said.

At the time he described it, Mr. Langlands believed it would take only a few years to formally prove the relationships of the fundamental lemma before mathematicians moved along toward proving other relationships encompassed by the Langlands Program.

Instead, Mr. Langlands spent more than a decade trying, calling the problem "more difficult than I thought." Mr. Ngo, who worked with Mr. Langlands at the Institute for Advanced Study before taking a new job at the University of Chicago, finally succeeded, last year, by taking a fundamentally new perspective to the problem that relied heavily on geometry, Mr. Frenkel said. In the process, Mr. Frenkel said, Mr. Ngo produced a series of important insights into mathematical theory that, among other things, should be helpful with further exploration of the mathematical and physical relationships encompassed by the overall Langlands Program.

Because of the close ties between mathematical theories and the real-world physical behaviors actually seen by scientists, the Langlands Program could open the way to using math to achieve a wider understanding of the physical world, down to the subatomic level, and possibly grasping key concepts that still elude physicists.

The $10-billion Large Hadron Collider project along the French-Swiss border is a spectacular high-profile and costly attempt to understand fundamental questions of physics by exploring the elemental particles that make up atoms. With paper and pencil, mathematicians such as Mr. Ngo and Mr. Langlands are tackling such questions from another perspective, largely using the tool of their mind.

Editor's note: In response to readers' questions, we have added another paragraph explaining Mr. Ngo's work on solving the Langlands fundamental lemma, along with a link to his proof (from Cornell University Library's arXiv.org, published in French).