# Richard Lawrence Taylor

### Quick Info

Born
19 May 1962
Cambridge, England

### Biography

Richard Taylor's parents are John Clayton Taylor and Gillian Mary Schofield. John Taylor is a mathematical physicist, now Emeritus Professor of Mathematical Physics at the University of Cambridge. Richard's mother, Mary Taylor, was a piano teacher. Richard was born in Cambridge but, when he was two years old, the family moved to Oxford where Richard was brought up and attended school. He attended Magdalen College School where he was taught mathematics by Tony Middleton who later became a Lecturer in Mathematics for Physics at Brasenose College, University of Oxford. Taylor said [12]:-
I became interested in mathematics very early, I suspect. My father is a theoretical physicist. There was always a culture of mathematical science in the family. I don't remember exactly, but certainly as a teenager I was interested in mathematics. I just enjoyed reading recreational books on mathematics and trying to do math problems and finding out about more advanced mathematics. There wasn't any one thing that struck me as particularly interesting. I guess already in high school it was clear that I was better than most of the other kids in mathematics.
He also wrote about his early experiences of mathematics in [11]:-
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.
After completing his school studies at Magdalen College School, Taylor returned to Cambridge where he matriculated at Clare College in 1980. He worked hard at his mathematical studies but also found time for other interests. He was president of 'The Archimedeans' in 1981 and 1982. This is a Cambridge mathematical society founded in 1935 which aims to promote cooperation between all Cambridge mathematical societies. He also enjoyed travelling, particularly to places where he could indulge his love of mountaineering. He visited the Alps, the Indian Himalayas and later the volcanoes of Ecuador and the large Karakoram mountain range. It was number theory which attracted him most among the mathematical topics he studied. He wrote [11]:-
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.
He was. however, somewhat unsure of his own abilities [12]:-
... as you go on, you're always mixing with people who are more talented in mathematics. It is never clear if you have a real talent or just appear talented in the group you are currently mixing with. I really enjoy mathematics. I think great interest in mathematics and determination to persevere accounts for more than people often give credit for. If you are very keen on working on mathematical problems, you usually get good at it, and I think this can make up for a fair amount of mathematical talent. I have certainly known people who are far brighter mathematicians than I am, but if they have thought about a problem for two days and can't solve it, they get bored with it and want to move on.
Taylor graduated from the University of Cambridge in 1984 and, after some doubts as to whether he was good enough to undertake research in an area as demanding as number theory, he decided that he would undertake graduate studies at Princeton in the United States. There he chose to work with Andrew Wiles who had taken up a post at the Institute for Advanced Study in Princeton in 1981, then was appointed a professor at Princeton University in the following year. Taylor spent four years at Princeton 1984-88, during which time he undertook research for a Ph.D. advised by Wiles. Taylor was awarded his Ph.D. in 1988 for his thesis On congruences between modular forms. Two papers coming from the work of his thesis appeared in 1989, namely On Galois representations associated to Hilbert modular forms, and Representations of Galois groups associated to Hilbert modular forms.

After graduating from Princeton, Taylor became a fellow of Clare College, Cambridge and a Royal Society European Exchange Fellowship funded a postdoctoral year 1988-89 at the Institut des Hautes Études Scientifiques outside Paris. One of the major attractions of returning to the University of Cambridge was the fact that John Coates, who had been Andrew Wiles' thesis advisor at Cambridge in the 1970s, had been appointed to the Sadleirian Chair of Mathematics at Cambridge in 1986. Taylor, still a fellow of Clare College, was appointed Assistant Lecturer (1989-92), Lecturer (1992-94), then Reader (1994-95) at Cambridge University during the six years 1989-95. Taylor moved to Oxford in 1995 when he was appointed Savilian Professor of Geometry. The announcement was as follows:-
Savilian Professorship of Geometry, Richard Lawrence Taylor, FRS (MA Cambridge, Ph.D. Princeton), Reader in Number Theory, University of Cambridge, has been appointed to the professorship with effect from 1 October 1995. Dr Taylor will be a fellow of New College.
Taylor married Christine Jiayou Chang in 1995. Christine Chang, who was an algebraist, had graduated from Harvard University and, in 1993, had been awarded a NSF Graduate Fellowships to study at the Massachusetts Institute of Technology.

Taylor wrote in [11]:-
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). ... 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.
As he explains in the above quote, after only one year at Oxford, Taylor moved to the United States when he was appointed as a Professor at Harvard University. He explained the reasons for his move in the interview [12] which he gave shortly after arriving in Harvard to take up the professorship:-
I guess I got the formal offer [from Harvard] in the spring [of 1996] from the dean, but we'd obviously talked about it with the faculty [at Harvard] for some time before that. One strong personal reason is that my wife's American and would like to be in America. Also it's a great department. Like I say, it's difficult to imagine a better collection of colleagues in my subject than there is here. By all accounts, the students here are very bright. I don't really have personal experience, but I'm sure it's true. I actually visited for six months a couple of years ago, and one thing I like is the sun. Somehow in Britain for half the year, it's extraordinarily dark. That's partly because it's further north and partly because there is more cloud cover. I've heard people complain that in the winter it's cold here, but at least you see the sun. And I like the energy; people are very energetic and enthusiastic here. Something I noticed is that in Britain it's cool to pretend you never do any work. Students there obviously do work because they learn the same stuff as anybody else, but they like to pretend they do nothing. Whereas [in the United States], people in Princeton would come to me and tell me they had spent the last twenty-four hours in the library. Here, they seem to pretend they work harder than they do. I suspect that people work the same in both places; it's just the gloss they put on it.
In 2002 Taylor was appointed as Herchel Smith Professor of Mathematics at Harvard. He continued to hold this role for the next ten years. From August 2010 to December 2011, he was a visitor at the Institute for Advanced Study at Princeton. In January 2012, he left his position at Harvard to be named the Robert and Luisa Fernholz Professor of Mathematics at the Institute for Advanced Study at Princeton. Taylor joined the Mathematics Department in the School of Humanities and Sciences of Stanford University in July 2018 when he was appointed as the new Barbara Kimball Browning Professor. This endowed chair is the highest honour that Stanford University can bestow on a faculty member.

For Taylor's 1996 reply to the question, "What are your Major Research Interests and Achievements?", see THIS LINK.

To understand the outstanding contributions which Taylor has continued to make we first list prizes and awards he has won and then give the citation for some of these. These awards include: London Mathematical Society Whitehead Prize (1990); the Ostrowski Prize (2001); the Fermat Prize (2001); the American Mathematical Society, Frank Nelson Cole Prize in Number Theory (2002); the Göttingen Academy of Sciences Dannie Heineman Prize (2005); the Shaw Prize in Mathematics (2007); the Clay Research Award (2007); and the Breakthrough Prize in Mathematics (2015). Other honours include: elected a fellow of the Royal Society (1995); elected a fellow of the American Mathematical Society (2012); elected to the National Academy of Sciences (2015); and elected to the American Philosophical Society (2018).

Let us now give some information about some of these prizes and the citation for Taylor.
The Ostrowski Prize.
The Ostrowski Foundation was created by Alexander Ostrowski, for many years a professor at the University of Basel. He left his entire estate to the foundation and stipulated that the income should provide a prize for outstanding recent achievements in pure mathematics and the foundations of numerical mathematics. The prize is awarded every other year.

The Ostrowski Prize: Citation for Richard Taylor.
Taylor has been a major contributor to some of the most spectacular developments in number theory over the last ten years. A particularly fascinating and rewarding theme in number theory has been the application of automorphic forms to arithmetic problems related to l-adic Galois representations. Taylor's extraordinary creativity and impressive technical command of both algebraic geometry and automorphic representation theory allowed him to make deep and profound discoveries in this area. He is best known for his input to the work of Andrew Wiles, proving the Taniyama-Shimura-Weil conjecture in sufficiently many cases to imply Fermat's Last Theorem. In a series of papers, jointly with Diamond, Conrad and Breuil, Taylor recently completed the proof of that conjecture: every rational elliptic curve is covered by a modular curve. The Taniyama-Shimura-Weil conjecture is an instance of the Global Langlands Program, relating automorphic representations and Galois representations. Another major achievement of Taylor, together with Michael Harris, is the proof of the Local Langlands Conjecture for $GL(n)$, which establishes a similar correspondence over the completions of $\mathbb{Q}$. A third and related series of papers, partly in collaboration with N Shepherd-Barron and K Buzzard, concerns a programme laid out by Taylor to prove the Artin Conjecture on the holomorphicity of $L$-functions of certain two-dimensional representations of the Galois group of the rational numbers.

The American Mathematical Society Cole Prize in Number Theory.
The Frank Nelson Cole Prize in Number Theory is awarded every three years for a notable research memoir in number theory that has appeared during the previous five years (until 2001, the prize was usually awarded every five years). The awarding of this prize alternates with the awarding of the Cole Prize in Algebra, also given every three years. These prizes were established in 1928 to honour Frank Nelson Cole on the occasion of his retirement as secretary of the American Mathematical Society after twenty-five years of service. He also served as editor-in-chief of the Bulletin for twenty-one years.

The Cole Prize in Number Theory: Citation for Richard Taylor.
The Frank Nelson Cole Prize in Number Theory is awarded to Richard Taylor of Harvard University for several outstanding advances in algebraic number theory. He led an effort to extend his earlier work with Wiles, to show that all elliptic curves over $\mathbb{Q}$ are modular, i.e., are factors of the Jacobians of modular curves. In his book with M Harris, he established the local Langlands conjecture, giving a complete parametrization of the $n$-dimensional representations of a Galois group of a local field. He has also made important progress on 2-dimensional Galois representations, establishing the Artin conjecture for an infinite class of nonsolvable cases, and increasing our understanding of the conjectures of Fontaine-Mazur and Serre.

The Shaw Prize in Mathematics.

The Breakthrough Prize in Mathematics.
The Breakthrough Prize in Mathematics was launched by Facebook founder Mark Zuckerberg and Russian entrepreneur Yuri Milner at the Breakthrough Prize ceremony in December 2014. It aims to recognize major advances in the field, honour the world's best mathematicians, support their future endeavours and communicate the excitement of mathematics to the general public.

The Breakthrough Prize: Citation for Richard Taylor.
The Breakthrough Prize is awarded to Richard Taylor, Institute for Advanced Study, for numerous breakthrough results in the theory of automorphic forms, including the Taniyama-Weil conjecture, the local Langlands conjecture for general linear groups, and the Sato-Tate conjecture.

In October-November 2013 Taylor delivered three lectures in the University of California Los Angeles Distinguished Lecture Series.
1. Title: Reciprocity laws.
Abstract:  Reciprocity laws provide a rule to count the number of solutions to a fixed polynomial equation, or system of polynomial equations, modulo a variable prime number. The rule will involve very different objects: automorphic forms and discrete subgroups of Lie groups. The prototypical example is Gauss' law of quadratic reciprocity, which concerns a quadratic equation in one variable. Another celebrated example is the Shimura-Taniyama conjecture which concerns a cubic equation in two variables. I will start with Gauss' law and work my way up to somewhat more complicated examples. At the end of the talk I hope to indicate the current state of our knowledge.
2. Title: Galois theory and locally symmetric manifolds.
Abstract:  I will describe recent results showing that one can attach Galois representations to classes in the cohomology of the certain locally symmetric (real) manifolds, namely the quotients of the space of totally positive real symmetric matrices by congruence subgroups of $GL(n,\mathbb{Z})$. I will discuss both my joint work with Harris, Lan and Thorne concerning cohomology with rational coefficients and the work of Scholze on cohomology with coefficients in a finite field. If time permits I will give some indication of the proofs.
3. Title: Reciprocity laws for regular, self-dual motive.
Abstract: I will discuss recent work with Stefan Patrikis proving the automorphy of regular self-dual motives over the rational numbers. In previous work with Barnet-Lamb, Gee, and Geraghty this was shown modulo an irreducibility hypothesis on the corresponding l-adic representations. The innovation in the more recent work is a simple trick that allows us to by-pass this irreducibility hypothesis that can be hard to check in practice.

### References (show)

1. 2002 Cole Prize in Number Theory, Notices Amer. Math. Soc. 49 (4) (2002), 476-478.
2. M Cimons, Unlocking the mysteries of mathematics, National Science Foundation (19 December 2014). https://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=133695&org=NSF&from=news
3. Iwaniec, Sarnak, and Taylor Receive Ostrowski Prize, Notices Amer. Math. Soc. 49 (7) (2002), 800-801.
4. Mathematician Taylor wins Shaw Prize, The Harvard Gazette (11 September 2007). https://news.harvard.edu/gazette/story/2007/09/mathematician-taylor-wins-shaw-prize/
5. Personal Profile of Dr Richard Lawrence Taylor, Department of Mathematics, Stanford University. https://www.msri.org/people/3960
6. Professor Richard Taylor, Department of Mathematics Stanford University (22 October 2018). https://mathematics.stanford.edu/2018/10/22/professor-richard-taylor-appointed-to-a-distinguished-professorship/
7. Richard L Taylor, National Academy of Sciences. http://www.nasonline.org/member-directory/members/20035967.html
8. Richard Taylor, School of Mathematics, Institute for Advanced Study. https://www.ias.edu/scholars/taylor
9. Richard Taylor, The Royal Society. https://royalsociety.org/people/richard-taylor-12396/
10. Richard Taylor, University of California Los Angeles Distinguished Lecture Series (October 2013). https://www.math.ucla.edu/dls/richard-taylor
11. The Shaw Prize in Mathematical Sciences 2007 (12 June 2007, Hong Kong). http://www.shawprize.org/en/shaw.php?tmp=3&twoid=50&threeid=60&fourid=88
12. S Sheffield, Exclusive Interview with Richard Taylor, Massachusetts Institute of Technology (1996). http://math.mit.edu/~sheffield/interview.html