Robert Phelan Langlands is an American-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 is an emeritus professor and occupies Albert Einstein's office at the Institute for Advanced Study in Princeton.
In mathematics, the modularity theorem 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.
There have been several attempts in history to reach a unified theory of mathematics. Some of the greatest mathematicians have expressed views that the whole subject should be fitted into one theory.
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.
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.
Carl Ludwig Siegel was a German mathematician specialising in number theory and celestial mechanics. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation and the Siegel mass formula for quadratic forms. He was named as one of the most important mathematicians of the 20th century.
Yutaka Taniyama was a Japanese mathematician known for the Taniyama–Shimura conjecture.
The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993 there has been a formal division into three categories.
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analogues of the roots of unity, as complex numbers that are particular values of the exponential function; the requirement is that such numbers should generate a whole family of further number fields that are analogues of the cyclotomic fields and their subfields.
In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field.
In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. The term "Shimura variety" applies to the higher-dimensional case, in the case of one-dimensional varieties one speaks of Shimura curves. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties.
In mathematics, the geometric Langlands correspondence is a geometric reformulation of the number theoretic Langlands correspondence.
Angus John Macintyre FRS, FRSE is a British mathematician and logician who is a leading figure in model theory, logic, and their applications in algebra, algebraic geometry, and number theory. He is Emeritus Professor of Mathematics, at Queen Mary University of London.
Fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets. A fuzzy subset A of a set X is a function A:X→L, where L is the interval [0,1]. This function is also called a membership function. A membership function is a generalization of a characteristic function or an indicator function of a subset defined for L = {0,1}. More generally, one can use a complete lattice L in a definition of a fuzzy subset A .
Tomio Kubota is a Japanese mathematician, who studies number theory. His contributions include works on p-adic L functions and real-analytic automorphic forms.
In number theory, the Eichler–Shimura congruence relation expresses the local L-function of a modular curve at a prime p in terms of the eigenvalues of Hecke operators. It was introduced by Eichler (1954) and generalized by Shimura (1958). Roughly speaking, it says that the correspondence on the modular curve inducing the Hecke operator Tp is congruent mod p to the sum of the Frobenius map Frob and its transpose Ver. In other words,
Michael Howard Harris is an American mathematician who deals with number theory and algebra. He made notable contributions to the Langlands program, for which he won the 2007 Clay Research Award. In particular, he, proved the local Langlands conjecture for GL(n) over a p-adic local field in, and was part of the team that proved the Sato–Tate conjecture.
In mathematics, Eichler cohomology is a cohomology theory for Fuchsian groups, introduced by Eichler (1957), that is a variation of group cohomology analogous to the image of the cohomology with compact support in the ordinary cohomology group. The Eichler–Shimura isomorphism, introduced by Eichler for complex cohomology and by Shimura (1959) for real cohomology, is an isomorphism between an Eichler cohomology group and a space of cusp forms. There are several variations of the Eichler–Shimura isomorphism, because one can use either real or complex coefficients, and can also use either Eichler cohomology or ordinary group cohomology as in. There is also a variation of the Eichler–Shimura isomorphisms using l-adic cohomology instead of real cohomology, which relates the coefficients of cusp forms to eigenvalues of Frobenius acting on these groups. Deligne (1971) used this to reduce the Ramanujan conjecture to the Weil conjectures that he later proved.
Haruzo Hida is a Japanese mathematician, known for his research in number theory, algebraic geometry, and modular forms.
