Richard Taylor | |
---|---|
![]() Taylor in 1999 | |
Born | Richard Lawrence Taylor 19 May 1962 Cambridge, England |
Nationality | British, American |
Alma mater | Clare College, Cambridge (BA) Princeton University (PhD) |
Spouse | Christine Taylor[ citation needed ] |
Awards | Whitehead Prize (1990) Fermat Prize (2001) Ostrowski Prize (2001) Cole Prize (2002) Shaw Prize (2007) Clay Research Award (2007) Breakthrough Prize in Mathematics (2015) |
Scientific career | |
Fields | Mathematics |
Institutions | University of Oxford Harvard University Institute for Advanced Study Stanford University |
Thesis | On congruences between modular forms (1988) |
Doctoral advisor | Andrew Wiles |
Doctoral students |
Richard Lawrence Taylor (born 19 May 1962) is a British [2] mathematician working in the field of number theory. [3] He is currently the Barbara Kimball Browning Professor in Humanities and Sciences at Stanford University. [4]
Taylor received the 2002 Cole Prize, the 2007 Shaw Prize with Robert Langlands, and the 2015 Breakthrough Prize in Mathematics.
He received his B.A. from Clare College, Cambridge. [5] [6] During his time at Cambridge, he was president of The Archimedeans in 1981 and 1982, following the resignation of his predecessor. [7] He earned his Ph.D. in mathematics from Princeton University in 1988 after completing a doctoral dissertation, titled "On congruences between modular forms", under the supervision of Andrew Wiles. [8]
He was an assistant lecturer, lecturer, and then reader at the University of Cambridge from 1988 to 1995. [9] From 1995 to 1996 he held the Savilian chair of geometry [5] at Oxford University and Fellow of New College, Oxford. [9] [6] He was a professor of mathematics at Harvard University from 1996 to 2012, at one point becoming the Herchel Smith Professor of Mathematics. [9] He moved to the Institute for Advanced Study as the Robert and Luisa Fernholz Professorship from 2012 to 2019. [9] He has been the Barbara Kimball Browning Professor in Humanities & Sciences at Stanford University since 2018. [4]
He served on the Mathematical Sciences jury for the Infosys Prize from 2012 to 2014.[ citation needed ]
One of the two papers containing the published proof of Fermat's Last Theorem is a joint work of Taylor and Andrew Wiles. [10]
In subsequent work, Taylor (along with Michael Harris) proved the local Langlands conjectures for GL(n) over a number field. [11] A simpler proof was suggested almost at the same time by Guy Henniart, [12] and ten years later by Peter Scholze.
Taylor, together with Christophe Breuil, Brian Conrad and Fred Diamond, completed the proof of the Taniyama–Shimura conjecture, by performing quite heavy technical computations in the case of additive reduction. [13]
In 2008, Taylor, following the ideas of Michael Harris and building on his joint work with Laurent Clozel, Michael Harris, and Nick Shepherd-Barron, announced a proof of the Sato–Tate conjecture, for elliptic curves with non-integral j-invariant. This partial proof of the Sato–Tate conjecture uses Wiles's theorem about modularity of semistable elliptic curves. [14]
He received the Whitehead Prize in 1990, the Fermat Prize and the Ostrowski Prize in 2001, the Cole Prize of the American Mathematical Society in 2002, and the Shaw Prize for Mathematics in 2007. [9] He received the 2015 Breakthrough Prize in Mathematics "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." [15]
He was elected a Fellow of the Royal Society in 1995. [9] In 2012 he became a fellow of the American Mathematical Society. [16] In 2015 he was inducted into the National Academy of Sciences. [17] He was elected to the American Philosophical Society in 2018. [18]
Taylor is the son of British physicist John C. Taylor. He is married and has two children. [19]
Sir Andrew John Wiles is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal and for which he was appointed a Knight Commander of the Order of the British Empire in 2000. In 2018, Wiles was appointed the first Regius Professor of Mathematics at Oxford. Wiles is also a 1997 MacArthur Fellow.
Robert Phelan Langlands, is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theory, for which he received the 2018 Abel Prize. He was an emeritus professor and occupied Albert Einstein's office at the Institute for Advanced Study in Princeton, until 2020 when he retired.
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and consequential conjectures about connections between number theory and geometry. Proposed by Robert Langlands, it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics."
The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.
Gorō Shimura was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem.
There have been several attempts in history to reach a unified theory of mathematics. Some of the most respected mathematicians in the academia have expressed views that the whole subject should be fitted into one theory.
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational numbers by reduction modulo almost all prime numbers p. Mikio Sato and John Tate independently posed the conjecture around 1960.
Ribet's theorem is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT). As shown by Serre and Ribet, the Taniyama–Shimura conjecture and the epsilon conjecture together imply that FLT is true.
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
Brian Conrad is an American mathematician and number theorist, working at Stanford University. Previously, he taught at the University of Michigan and at Columbia University.
A modular elliptic curve is an elliptic curve E that admits a parametrisation X0(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.
In mathematics, the local Langlands conjectures, introduced by Robert Langlands, are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G over a local field F, and representations of the Langlands group of F into the L-group of G. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups.
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using current knowledge by almost all contemporary mathematicians.
Michael Howard Harris is an American mathematician known for his work in number theory. He is a professor of mathematics at Columbia University and professor emeritus of mathematics at Université Paris Cité.
Guy Henniart (born 1953, Santes) is a French mathematician at Paris-Sud 11 University. He is known for his contributions to the Langlands program, in particular his proof of the local Langlands conjecture for GL(n) over a p-adic local field—independently from Michael Harris and Richard Taylor—in 2000.
Haruzo Hida is a Japanese mathematician, known for his research in number theory, algebraic geometry, and modular forms.
Laurent Clozel is a French mathematician and professor at Paris-Saclay University. His mathematical work is in the area of automorphic forms, including the Langlands program.
Jack A. Thorne is a British mathematician working in number theory and arithmetic aspects of the Langlands Program. He specialises in algebraic number theory.
Sug Woo Shin is a professor of mathematics at the University of California, Berkeley working in number theory, automorphic forms, and the Langlands program.