query: field_of_work fred diamond
supports: In mathematics, a rational number is any number that can be expressed as the quotient or fraction "p"/"q" of two integers, a numerator "p" and a non-zero denominator "q". Since "q" may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", is usually denoted by a boldface Q (or blackboard bold formula_1, Unicode ); it was thus denoted in 1895 by Giuseppe Peano after "quoziente", Italian for "quotient". In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. Fred Irvin Diamond ( born November 19 , 1964 ) is a mathematician , known for his role in proving the modularity theorem for elliptic curves . His research interest is in modular forms and Galois representations . Diamond received his B.A. from the University of Michigan in 1983 , and received his Ph.D. in mathematics from Princeton University in 1988 as a doctoral student of Andrew Wiles . He has held positions at Brandeis University and Rutgers University , and is currently a professor at King 's College London . Diamond is the author of several research papers , and is also a coauthor along with Jerry Shurman of A First Course in Modular Forms , in the Graduate Texts in Mathematics series published by Springer - Verlag . Robert Phelan Langlands (born October 6, 1936) 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. He is an emeritus professor and occupies Albert Einstein's office at the Institute for Advanced Study in Princeton. In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theoremsand generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally "deductive", in contrast to the notion of a scientific law, which is "experimental". Mathematics (from Greek  "máthma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Christophe Breuil (born 1968) is a French mathematician, who works in algebraic geometry and number theory. A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's problems.<br>
  It can also be a problem referring to the nature of mathematics itself, such as Russell's Paradox. In mathematics, the Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by . In mathematics, the modularity theorem (formerly called the TaniyamaShimuraWeil conjecture and several related names) states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's last theorem. Later, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor extended Wiles' techniques to prove the full modularity theorem in 2001. The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved. However, proved that elliptic curves defined over real quadratic fields are modular. Brian Conrad (born November 20, 1970), is an American mathematician and number theorist, working at Stanford University. Previously he taught at the University of Michigan and at Columbia University. Sir Andrew John Wiles (born 11 April 1953) is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is most notable for proving Fermat's Last Theorem, for which he received the 2016 Abel Prize. Wiles has received numerous other honours. In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been known to have infinitely many solutions since antiquity. In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group "G" to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup formula_1 of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. A mathematician is someone who uses an extensive knowledge of mathematics in his/her work, typically to solve mathematical problems. In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form
candidates: action computing french geometry harmonic analysis law mathematician mathematics number theory research topology
answer: number theory