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."
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.
Ilya Piatetski-Shapiro was a Soviet-born Israeli mathematician. During a career that spanned 60 years he made major contributions to applied science as well as pure mathematics. In his last forty years his research focused on pure mathematics; in particular, analytic number theory, group representations and algebraic geometry. His main contribution and impact was in the area of automorphic forms and L-functions.
In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group G on the space L2(Γ\G) of square-integrable functions, where Γ is a cofinite discrete group. The character is given by the trace of certain functions on G.
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering.
In mathematics, a Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory.
Alexandre Aleksandrovich Kirillov is a Soviet and Russian mathematician, known for his works in the fields of representation theory, topological groups and Lie groups. In particular he introduced the orbit method into representation theory. He is an emeritus professor at the University of Pennsylvania.
Alexander Alexandrovich Kirillov Jr. is a Russian-born American mathematician, working in the area of representation theory and Lie groups. He is a son of Russian mathematician Alexandre Kirillov.
Georgi Evgen'evich Shilov was a Soviet mathematician and expert in the field of functional analysis, who contributed to the theory of normed rings and generalized functions.
Naum Yakovlevich Vilenkin was a Soviet mathematician, an expert in representation theory, the theory of special functions, functional analysis, and combinatorics. He is best known as the author of many books in recreational mathematics aimed at middle and high school students.
Victor Ginzburg is a Russian American mathematician who works in representation theory and in noncommutative geometry. He is known for his contributions to geometric representation theory, especially, for his works on representations of quantum groups and Hecke algebras, and on the geometric Langlands program. He is currently a Professor of Mathematics at the University of Chicago.
In the mathematical field of functional analysis, a Gelfand–Shilov space is a space of test functions for the theory of generalized functions, introduced by Gelfand and Shilov.
Andrei Vladlenovich Zelevinsky was a Russian-American mathematician who made important contributions to algebra, combinatorics, and representation theory, among other areas.
Sergey Vladimirovich Fomin is a Russian American mathematician who has made important contributions in combinatorics and its relations with algebra, geometry, and representation theory. Together with Andrei Zelevinsky, he introduced cluster algebras.
In mathematics, a Siegel domain or Piatetski-Shapiro domain is a special open subset of complex affine space generalizing the Siegel upper half plane studied by Siegel (1939). They were introduced by Piatetski-Shapiro in his study of bounded homogeneous domains.
Vladimir Solomonovich Retakh is a Russian-American mathematician who made important contributions to Noncommutative algebra and combinatorics among other areas.
Mark Iosifovich Graev was a Russian mathematician. He is known as one of the namesakes in the Gelfand–Graev representation.
Mikhail Kapranov, is a Russian mathematician, specializing in algebraic geometry, representation theory, mathematical physics, and category theory. He is currently a professor of the Kavli Institute for the Physics and Mathematics of the Universe at the University of Tokyo.
Dmitrii Abramovich Raikov was a Russian mathematician who studied functional analysis.
