field_of_work fred diamond
In ___MASK2___ , 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 ___MASK90___ 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 ___MASK17___ . Modular forms appear in other areas , such as algebraic topology , sphere packing , and string theory . In ___MASK2___ , a theorem is a statement that has been proved on the basis of previously established statements , such as other theorems  and 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 ___MASK66___ , which is " experimental ". ___MASK2___ ( 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 ___MASK2___ . In ___MASK2___ , an elliptic curve is a plane algebraic curve defined by an equation of the form Brian Conrad ( born November 20 , 1970 ), is an American ___MASK94___ and number theorist , working at Stanford University . Previously he taught at the University of Michigan and at Columbia University . Christophe Breuil ( born 1968 ) is a ___MASK64___ mathematician , who works in algebraic geometry and ___MASK17___ . A ___MASK94___ is someone who uses an extensive knowledge of ___MASK2___ in his / her work , typically to solve mathematical problems . In ___MASK17___ , 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 ___MASK2___ , the Langlands program is a web of far - reaching and influential conjectures that relate Galois groups in algebraic ___MASK17___ to automorphic forms and representation theory of algebraic groups over local fields and adeles . It was proposed by . In ___MASK80___ and ___MASK17___ , 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 . In ___MASK2___ , the modularity theorem ( formerly called the Taniyama  Shimura  Weil 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 ___MASK23___ , 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 . Fred Irvin Diamond ( born November 19 , 1964 ) is a ___MASK94___ , known for his role in proving the modularity theorem for elliptic curves . His ___MASK29___ 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 ___MASK29___ 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 . In ___MASK2___ , 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 ". Sir Andrew John Wiles ( born 11 April 1953 ) is a British ___MASK94___ and a Royal Society ___MASK29___ Professor at the University of Oxford , specialising in ___MASK17___ . He is most notable for proving Fermat ' s Last Theorem , for which he received the 2016 Abel Prize . Wiles has received numerous other honours . A mathematical problem is a problem that is amenable to being represented , analyzed , and possibly solved , with the methods of ___MASK2___ . This can be a real - world problem , such as ___MASK34___ the orbits of the planets in the solar system , or a problem of a more abstract ___MASK75___ , such as Hilbert ' s problems .< br > It can also be a problem referring to the ___MASK75___ of ___MASK2___ itself , such as Russell ' s Paradox . Robert Phelan Langlands ( born October 6 , 1936 ) is a Canadian ___MASK94___ . 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 ___MASK17___ . He is an emeritus professor and occupies Albert Einstein ' s office at the Institute for Advanced Study in Princeton .