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Some mathematicians are immortalised by a theorem. Others by a conjecture.
But of the great mathematicians only Robert P Langlands is immortalised by a program.
Today, Langlands, aged 81, was awarded the 2018 Abel Prize, one of the most prestigious awards in mathematics, for work that has become known as the Langlands program, an ambitious project that is often called a ‘grand unified theory of mathematics.’
The prize is awarded by the Norwegian Academy of Science and Letters and is worth 6 million Norwegian krone (about £550,000). It is considered by many to be a maths equivalent of the Nobel Prize, which has no prize for mathematics.
I wrote the following short biography of Langlands, and a short article explaining some of the concepts in the prize citation, for the Norwegian Academy, reprinted below here:
A biography of Robert Langlands
In January 1967, Robert Langlands, a 30-year-old associate professor at Princeton, wrote a letter to the great French mathematician André Weil, aged 60, outlining some of his new mathematical insights.
“If you are willing to read it as pure speculation I would appreciate that,” he wrote. “If not – I am sure you have a waste basket handy.”
Langlands’ modesty now reads like an almost comic piece of understatement. His 17-page letter introduced a theory that created a whole new way of thinking about mathematics: it suggested deep links between two areas, number theory and harmonic analysis, that had previously been considered unrelated.
In fact, so radical were his insights, and so rich the mechanisms he suggested to bridge these mathematical elds, that his letter began a project, the Langlands program, that has enlisted hundreds of the world’s best mathematicians over the last fifty years. No other project in modern mathematics has as wide a scope, has produced so many deep results, and has so many people working on it. As its depth and breadth have grown, the Langlands program is frequently described as a grand unifed theory of mathematics.
Robert Phelan Langlands was born in New Westminster, Greater Vancouver, Canada, in 1936. When he was nine, he moved to a small tourist town near the US border where his parents had a shop selling building supply materials. He had no intention of going to university until a teacher told him, in front of his classmates, that it would be a betrayal of his God-given talents.
Langlands enrolled at the University of British Columbia aged 16. He completed his bachelor’s degree in mathematics in 1957, and his master’s degree a year later. He moved to Yale University for his doctorate, completing his PhD thesis, Semi-groups and representations of Lie groups, in his first year there. In his second year he began to study the work of the Norwegian Atle Selberg, which later became central to his own research.
In 1960, Langlands joined Princeton University as an instructor, where he rubbed shoulders with Selberg, as well as André Weil and Harish-Chandra, all of whom were at the nearby Institute for Advanced Study. He was especially influenced by the work of Harish-Chandra on automorphic forms. Langlands was also learning other areas of mathematics, such as class field theory, an area he was nudged into by his colleague Salomon Bochner, who encouraged him to give a course in it. In 1962, Langlands was appointed a member in the Institute’s School of Mathematics.
During the Christmas break of 1966, Langlands came up with the basic idea of “functoriality”, a mechanism for linking ideas in number theory to those in automorphic forms. He bumped into Weil in a corridor in the beginning of January 1967 and began to explain his discovery. Weil suggested he write up his thoughts in a letter.
Langlands swiftly wrote the letter in longhand. Weil had the letter typed up and it was widely circulated among Mathematicians. Over the next few years, the letter provided many of them with a number of new, deep and interesting problems and as more people joined the project to prove his conjectures the enterprise became known as the Langlands program “There were some ne points that were right that rather surprise me to this day,” Langlands later said about the letter. “There was evidence that these L-functions were good but that they would have these consequences for algebraic number theory was by no means certain.”
Langlands spent the year 1967-68 at the Middle East Technical University in Ankara. He speaks uent Turkish. An enthusiastic learner of languages, he also speaks German and Russian.
Langlands returned to Yale where he developed his twin ideas of functoriality and reciprocity and published them in Problems in the Theory of Automorphic Forms (1970). In 1972 he returned to Princeton as a professor at the Institute for Advanced Study, where he has been ever since.
Throughout the 1970s, Langlands continued to work on ideas within his programme. In the mid-1980s, he turned his attention to percolation and conformal invariance, problems from theoretical physics. In recent years he has been looking back at ideas that he pioneered, such as one called “endoscopy”.
Langlands has won many awards, including the rst US National Academy of Sciences Award in Mathematics in 1988 “for his extraordinary vision”. He shared the 1996 Wolf Prize with Andrew Wiles for his “path blazing work”. Other awards include the 2005 American Mathematical Society Steele Prize, the 2006 Nemmers Prize in Mathematics and the 2007 Shaw Prize in Mathematical Sciences (with Richard Taylor).
While at UBC, aged 19, he married Charlotte Lorraine Cheverie. He has four children with Charlotte, and several grandchildren.
At the age of 81, he continues to work at the Institute for Advanced Study, where he is now Emeritus Professor, and where he occupies the office once used by Albert Einstein.
A glimpse of the laureate’s work
The citation for Robert P. Langlands’ Abel Prize begins:
The Langlands program predicts the existence of a tight web of connections between automorphic forms and Galois groups.
In order to understand the significance of the Langlands program we need to consider the mathematical histories of both of these concepts: automorphic forms and Galois groups.
Harmonic analysis
Automorphic forms come from an area of mathematics called harmonic analysis, which can be thought of, in very approximate and general terms, as the study of periodic waves. The best-known periodic wave is the sine wave, which consists of a peak and a trough repeating ad infinitum. The use of the word harmonic here comes from the sine wave’s application to the physics of sound. A musical note, for example, a C played by a violin, can be understood as the superposition of many sine waves, each of these waves representing a “harmonic” of the C. Automorphic forms are a generalization of the idea of periodic waves, expressed using a more sophisticated geometrical language.
Number theory
Galois groups are a concept that emerged from number theory, the study of the properties of numbers. One important topic within number theory is how to solve polynomial equations, meaning equations where the exponents are positive whole numbers. For example, how do we solve the following equation?
x2 + x + 1 = 0
This polynomial is a quadratic equation and its two solutions be calculated using the quadratic formula known to most schoolchildren. The solutions are
x = –½ + (√–3)/2
x = –½ – (√–3)/2
The solutions look similar: the first part in both cases is –½, and the second is either plus or minus (√–3)/2. In other words, they display a symmetry: you just flip the sign from plus to minus or from minus to plus to get from one to the other. In the early nineteenth century, the French mathematician Evariste Galois studied the symmetries between solutions of polynomial equations. The table of these symmetrical relations is now called a Galois group.
The connections
The Langlands program is a hugely ambitious project to bridge harmonic analysis and number theory, initially by showing deep connections between automorphic forms and Galois groups. Harmonic analysis and number theory are two separate fields, each with their own concepts, techniques and terminology; the program reveals powerful equivalences between them.
One basic concept crucial to Langlands’ ideas is modular arithmetic, a way of doing arithmetic with a fixed set of consecutive numbers such that when you count beyond the top you start from zero again. One example of modular arithmetic is the 12-hour clock. If it is 11 o’clock and you add 5 hours, using normal arithmetic you would get to 16 o’clock. But there is no 16 in the 12 -hour clock! We all know that 11 o’clock plus 5 hours is 4 o’clock, since once you hit 12 you start from zero again.
In his Disquisitiones Arithmeticae (1801), the German mathematician Carl Friedrich Gauss established a theory of modular arithmetic and presented as its “fundamental theorem” the law of quadratic reciprocity, which is about the solvability of quadratic equations using modular arithmetic. Let’s look again at the quadratic equation above x2 + x + 1 = 0. If we are considering modular arithmetic with a modulus of 3, meaning we are using only the three numbers 0, 1 and 2, then this equation has a solution of x = 1, since 12 + 1 + 1 = 3, which is the same as 0 when the modulus is 3. Since Gauss, mathematicians have been interested in how the solvability of certain types of equation depends on the modulus and how this relates to the Galois groups of these equations.
A specific case of how the Langlands program connects number theory and harmonic analysis can be seen by considering a type of polynomial equation called an “elliptic curve”. If you take an elliptic curve and find the number of solutions it has for every modulus when the modulus is a prime number (that is, the numbers 2, 3, 5, 7, 11, … , which are those numbers only divisible by themselves and 1) you will generate a sequence of numbers. This sequence of numbers, however, can also be generated by a different type of mathematical object that is (very approximately) analogous to a periodic wave, and can be investigated using the tools of harmonic analysis.
In his 1967 letter to André Weil and in his 1970 Problems in the Theory of Automorphic Forms, Langlands made many wide-reaching conjectures that link number theory and harmonic analysis, which most experts believe are true but many of which have not yet been proved. Even so, it has been one of the most fertile areas for mathematical research. In 2002 and 2010 mathematicians were awarded the Fields Medal for proving Langlands’ conjectures.
The Langlands program is exciting for mathematicians because it bridges apparently unrelated disciplines, revealing a deeper structure underlying all mathematics and providing new ways to solve intractable problems. But it is also intoxicating because of the nature of the connections: number theory is an area where numbers often appear with no predictable order, yet automorphic forms are full of smooth curves, regular patterns and beautiful symmetries.
(GUARDIAN)
