# Richard Taylor wins 2007 Shaw Prize

In 2007 Robert Langlands and Richard Taylor were awarded, in equal shares, the Shaw Prize, "for initiating and developing a grand unifying vision of mathematics
that connects prime numbers with symmetry." We present below the Biographical Note for Richard Taylor and Richard Taylor's autobiography written as Shaw Prize winner. We also give the Press announcement and the essay about Robert Langlands and Richard Taylor's contributions. We have included the description of Langlands work simply because it is so interlinked with that of Richard Taylor.
1. Biographical Note for Richard Taylor

Richard Taylor, born 1962 is currently the Herchel Smith Professor of Mathematics at Harvard University, a post he has held since 2002. Professor Taylor was born in England. He received his BA from Cambridge University in 1984 and his PhD from Princeton University 4 years later. He taught at Cambridge University from 1989 to 1995 and held the Savilian Chair of Geometry at Oxford University from 1995 to 1996. He is a Fellow of the Royal Society of London.

12 June 2007, Hong Kong

2. Richard Taylor's autobiography

I was born on May 19, 1962 in Cambridge, England, but two years later we moved to Oxford where I spent the rest of my childhood. My mother, Mary, was a piano teacher and my father, John, a theoretical physicist. I enjoyed mathematics from a young age and was blessed with a number of inspiring mathematics teachers, including Tony Middleton at Magdalen College School. Although never a star at them, I greatly enjoyed the mathematics olympiads, which gave me my first experience of working on problems which took more than a few minutes to solve. But the biggest influence on my early scientific development was undoubtedly my father, who taught me never to be satisfied until I had really understood something completely. I also learnt from him not to fear asking simple-minded questions.

I was an undergraduate at Clare College, Cambridge. At this stage I developed a passion for travel and mountaineering, visiting the Alps, the Indian Himalayas and later the Karakoram and Ecuadorean volcanoes. I found it a great way to relax from mathematics which otherwise could be very consuming. It also became clear to me that number theory was the field that I found most exciting. I was attracted by the combination of simple problems, beautiful structure and the variety of techniques that were employed. However I very nearly chose to do graduate work in another area because I felt my abilities were insufficient to make an impact in such a hard field with so many outstanding practitioners. I overcame these doubts and went to graduate school in Princeton. Here I chose to work with Andrew Wiles attracted both by the beauty of his work and his approachability. It was a wise choice. Andrew's influence on my work has been enormous. After completing my PhD I spent a year at the Institut des Hautes Etudes Scientifiques outside Paris before returning to Cambridge University and Clare College. I stayed there for the next 6 years, during which time I benefited greatly from John Coates' support.

In 1994 I had the wonderful good fortune to meet Christine Chang, who has made my life much happier. We married in August 1995 and now have two children: Jeremy (born in 1998) and Chloe (born in 2000). Since then I have devoted significantly less time to mathematics, but paradoxically my mathematical work has improved.

In an effort to combine our two scientific careers I left Cambridge University following my marriage to Christine, first for the Savilian chair of geometry at Oxford and then a year later for Harvard University, where I am currently the Herchel Smith professor of mathematics. At Harvard I have found a supportive and stimulating home with incomparable colleagues and students.

My mathematical interests centre on the relationship between two very different kinds of symmetry: certain discrete symmetries of polynomial equations discovered by Galois in the first half of the 19th century, and other continuous symmetries arising in geometry. In the simplest (commutative) case this relationship was one of the great mathematical achievements of the first half of the 20th century (class field theory). More recently a much more general (non-commutative) theory has developed which is often loosely described as the Langlands program. I am fascinated by the way these ideas relate two very different kinds of mathematics (one coming from algebra, the other more closely related to analysis) which on the face of it have no reason to be related. I am also deeply impressed at how progress on this "program" has led to the solution of old, concrete problems in number theory. The most notable, but certainly not the only instance of this, being Andrew Wiles' proof of Fermat's last theorem. (It was a wonderful opportunity when in December 1993 Andrew asked me to help him repair the gap in his first attempt to prove Fermat's last theorem, a task at which we succeeded in less than a year, though we used a wholly unexpected argument.)

Class field theory can be considered the one dimensional case of the program. My early work is concerned with the two dimensional case, most notably my work with Wiles alluded to above and its continuation with Breuil, Conrad and Diamond to prove the full Shimura-Taniyama conjecture which has important applications to the arithmetic of elliptic curves. Also my discovery of potential modularity results, which led to the proof of the meromorphic continuation and functional equation of the L-functions of all regular rank two motives. Subsequently Khare and Wintenberger again made use of these ideas in their ground-breaking proof of Serre's conjecture. More recently I turned my attention to any number of dimensions, most often in a long collaboration with Michael Harris. I consider our proof of the local Langlands conjecture and our work (in part with Clozel and Shepherd-Barron) on modularity lifting theorems and potential modularity theorems in any number of dimensions to be the highlights of this work. An application of this is the proof of the Sato-Tate conjecture (for elliptic curves with non-integral j-invariant). This account will make clear that I am someone who works best in collaboration with others. I am extremely fortunate to have had fruitful and very enjoyable collaborations with many different colleagues. I am also blessed to have been able to work with 20 very talented PhD students.

11 September 2007, Hong Kong

3. Richard Taylor - Press Release

The Shaw Prize in Mathematical Sciences 2007 will be awarded in equal shares to Robert Langlands and Richard Taylor. Richard Taylor has made many extraordinary contributions to modern number theory, and more specifically to the framework of the Langlands program, where he has, in recent years, solved several important problems that had been long-standing conjectures.

Mathematical Sciences Selection Committee
The Shaw Prize
11 September 2007, Hong Kong

4. Robert Langlands and Richard Taylor - The essay

The work of Robert Langlands and Richard Taylor, taken together, provides us with an extraordinary unifying vision of mathematics. This vision begins with "Reciprocity", the fundamental pillar of arithmetic of previous centuries, the legacy of Gauss and Hilbert. Langlands had the insight to imbed Reciprocity into a vast web of relationships previously unimagined. Langlands' framework has shaped - and will continue to shape, unify, and advance - some of the most important research programmes in the arithmetic of our time as well as the representation theory of our time. The work of Taylor has, by a route as successful as it is illuminating, established - in the recent past - various aspects of the Langlands programme that have profound implications for the solution of important open problems in number theory.

For a prime number p form the (seemingly elementary) function that associates to an integer n the value +1 if n is a square modulo p, the value -1 if it isn't, and the value 0 if it is divisible by p. It was surely part of Langlands' initial vision that such functions and their number theory might be relatively faithful guides to the vast number-theoretic structure concealed in the panoply of automorphic forms associated to general algebraic groups. Langlands, viewing automorphic forms as certain kinds of representations (usually infinite-dimensional) of algebraic groups, discovered a unification of the two subjects, number theory and representation theory, that has provided mathematics with the astounding dictionary it now is in the process of developing and applying. Namely, the Langlands Philosophy: a dictionary between number theory and representation theory which has the uncanny feature that many elementary representation-theoretic relationships become - after translation by this dictionary – profound, and otherwise unguessed, relationships in number theory, and conversely.

In the mid 1960's Robert Langlands was one of the prime movers in the development of the general analytic theory of automorphic forms and their relationship to representation theory. Of particular note is his much celebrated general theory of Eisenstein series. Remarkably quickly after this, he was able to enunciate in a rather precise way the audacious "Langlands philosophy" which has guided the subject ever since. This includes his extremely general "reciprocity conjecture" connecting automorphic forms with number theory and his "principle of functoriality", a beautiful conjecture that subsumes all these ideas in terms of internal properties of representations. In the 1970's and 1980's Langlands went on to attack many important special cases of his conjectures using generalisations of the Selberg Trace Formula. Of particular note is his theory of cyclic base change for $GL(2)$, an example of "functoriality" which has profound applications to number theory. He pioneered the use of the trace formula to study Shimura varieties. He also laid out a very detailed blueprint (the theory of "endoscopy") on how to overcome deep problems that were encountered when trying to apply the trace formula to analyse Shimura varieties or to prove cases of functoriality. In sum, Langlands' insight offers us a grand unification, already used to establish some of the deepest advances in number theory in recent years.

Indeed, it is thanks to the work of Richard Taylor that we now have some of these advances. To cite the most recent of these breakthroughs, he and co-workers (Michael Harris, Laurent Clozel, and Nicholas Shepherd-Barron) have established an important part of a basic conjecture that has been around for 40 years. At the same time, they have extended - in a striking way - our ability to make use of Langlands' ideas, in combination with work of others, for arithmetic purposes. The technical statement of what they have done is to have proved the Sato-Tate conjecture for elliptic curves over totally real fields, provided that the curve has a place of multiplicative reduction. The Sato-Tate conjecture predicts that certain error terms in a broad class of important numerical functions of prime numbers conform to a specific probability distribution. In this recent work we see otherwise separate mathematical sub-disciplines coming together and connecting with each other in an illuminating way. Moreover, the successful strategy adopted, in keeping with Langlands' principle of functoriality, involves an infinite sequence of automorphic forms attached to algebraic groups of higher and higher rank. All this is surely just the beginning of a much bigger story, as envisaged by Langlands.

Richard Taylor's earlier work includes his celebrated collaboration with Wiles on the resolution of Fermat's Last Theorem followed by his quite significant contribution to the collaborative effort to finish fully the modularity of elliptic curves over the rational number field, his collaboration with Michael Harris culminating in the resolution of the local Langlands' Conjecture for the general linear group in $n$ dimensions, and his work resolving the classical Artin conjecture for a quite important class of non-solvable Galois representations of degree two.

The work of Robert Langlands and Richard Taylor demonstrates the profundity and the vigour of modern number theory and representation theory. Together they amply deserve the honour of the Shaw Prize.

Mathematical Sciences Selection Committee
The Shaw Prize
11 September 2007, Hong Kong

Last Updated November 2019