Joe Peter Buhler (born 1950 in Vancouver, Washington) is an American mathematician known for his contributions to algebraic number theory, algebra and cryptography.
Buhler received his undergraduate degree from Reed College in 1972, and his Ph.D. from Harvard University in 1977 with thesis Icosahedral Galois Representations and thesis advisor John Tate. [1] [2] Buhler was a professor at Reed College in Portland, Oregon from 1980 until his retirement in 2005. [3] From 2004 to 2017, he was director of the IDA Center for Communications Research in La Jolla, California. [4]
In 1997, he introduced, with Zinovy Reichstein, the concept of essential dimension. [5]
Buhler is involved in a project to numerically verify the Kummer–Vandiver conjecture of Harry Vandiver and Ernst Eduard Kummer concerning the class number of cyclotomic fields. Vandiver proved it with a desk calculator up to class number 600, Derrick Lehmer (in the late 1940s) to about 5000, and Buhler with colleagues (in 2001) to 12 million. [6] He continues the project with David Harvey and others. [7]
He was elected a Fellow of the American Mathematical Society in 2012.
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential 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."
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group. Every finite extension of a finite field is a cyclic extension.
Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003.
Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject.
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers.
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa (1959), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently, Ralph Greenberg has proposed an Iwasawa theory for motives.
In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.
In mathematics, the ADE classification is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in. The complete list of simply laced Dynkin diagrams comprises
In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime p does not divide the class number hK of the maximal real subfield of the p-th cyclotomic field. The conjecture was first made by Ernst Kummer on 28 December 1849 and 24 April 1853 in letters to Leopold Kronecker, reprinted in, and independently rediscovered around 1920 by Philipp Furtwängler and Harry Vandiver,
Nathan Jacobson was an American mathematician.
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and the Langlands program. So far, only a small part of such a theory has been put on a firm basis.
In mathematics, essential dimension is an invariant defined for certain algebraic structures such as algebraic groups and quadratic forms. It was introduced by J. Buhler and Z. Reichstein and in its most generality defined by A. Merkurjev.
In number theory, a compatible system of ℓ-adic representations is an abstraction of certain important families of ℓ-adic Galois representations, indexed by prime numbers ℓ, that have compatibility properties for almost all ℓ.
In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic. For example, the domain could be the p-adic integersZp, a profinite p-group, or a p-adic family of Galois representations, and the image could be the p-adic numbersQp or its algebraic closure.
In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles (1984). The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers.
Zinovy Reichstein is a Russian-born American mathematician. He is a professor at the University of British Columbia in Vancouver. He studies mainly algebra, algebraic geometry and algebraic groups. He introduced the concept of essential dimension.
Aleksandr Sergeyevich Merkurjev is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev is a professor at the University of California, Los Angeles.
In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic forms on these groups and representations of Galois groups.
In mathematics, base change lifting is a method of constructing new automorphic forms from old ones, that corresponds in Langlands philosophy to the operation of restricting a representation of a Galois group to a subgroup.