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D-modules
In mathematics, a ''D''-module is a module over a ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Since around 1970, ''D''-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of Masaki Kashiwara. The methods of ''D''-module theory have always been drawn from sheaf theory and other techniques with inspiration from the work of Alexander Grothendieck in algebraic geometry. This approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators. The strongest results are obtained for over-determined systems ( holonomic systems), and on the characteristic variety cut out by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Hilbert Correspondence
In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generalizations of this. The original setting appearing in Hilbert's twenty-first problem was for the Riemann sphere, where it was about the existence of systems of linear regular differential equations with prescribed monodromy representations. First the Riemann sphere may be replaced by an arbitrary Riemann surface and then, in higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension > 1. There is a correspondence between certain systems of partial differential equations (linear and having very special properties for their solutions) and possible monodromies of their solutions. Such a result was proved for algebraic connections with regular singularities by Pierre Deligne (1970, generalizing existing work in the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernstein–Sato Polynomial
In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by and , . It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related to the Bernstein polynomials used in approximation theory. It has applications to singularity theory, monodromy theory, and quantum field theory. gives an elementary introduction, while and give more advanced accounts. Definition and properties If f(x) is a polynomial in several variables, then there is a non-zero polynomial b(s) and a differential operator P(s) with polynomial coefficients such that :P(s)f(x)^ = b(s)f(x)^s. The Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such polynomials b(s). Its existence can be shown using the notion of holonomic D-modules. proved that all roots of the Bernstein–Sato polynomial are negative rational numbers. The Bernstein–Sato polynomial can also be de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Bernstein
Joseph Bernstein (sometimes spelled I. N. Bernshtein; ; ; born 18 April 1945) is a Soviet-born Israeli mathematician working at Tel Aviv University. He works in algebraic geometry, representation theory, and number theory. Biography Bernstein received his Ph.D. in 1972 under Israel Gelfand at Moscow State University. In 1981, he emigrated to the United States due to growing antisemitism in the Soviet Union. Bernstein was a professor at Harvard during 1983-1993. He was a visiting scholar at the Institute for Advanced Study in 1985-86 and again in 1997-98. In 1993, he moved to Israel to take a professorship at Tel Aviv University (emeritus since 2014). Awards and honors Bernstein received a gold medal at the 1962 International Mathematical Olympiad. He was elected to the Israel Academy of Sciences and Humanities in 2002 and was elected to the United States National Academy of Sciences in 2004. In 2004, Bernstein was awarded the Israel Prize for mathematics. In 1998, he was an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Masaki Kashiwara
is a Japanese mathematician and professor at the Kyoto University Institute for Advanced Study (KUIAS). He is known for his contributions to algebraic analysis, microlocal analysis, ''D''-module theory, Hodge theory, sheaf theory and representation theory. He was awarded the Abel Prize in 2025, and is the award's first recipient from Japan. Biography Kashiwara was born in Yūki, Ibaraki on January 30, 1947. One of his early mathematical fascinations was the tsurukamezan problem, which asks the number of cranes and turtles given a set number of legs and heads. Kashiwara spent his undergraduate years at the University of Tokyo (UTokyo), earning his bachelor's degree in mathematics in 1969. He then went on to study at the same institution for his master's degree, which he completed in 1971. At UTokyo, Kashiwara was a student of Mikio Sato. His master's thesis, written in Japanese, laid the foundations for the study of D-modules. He continued studying under Sato at Kyoto Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zoghman Mebkhout
Zoghman Mebkhout (born 1949 ) (زغمان مبخوت) is a France, French-Algerian mathematician. He is known for his work in algebraic analysis, geometry and representation theory, more precisely on the theory of D-module, ''D''-modules. Career Mebkhout is currently a research director at the French National Centre for Scientific Research and in 2002 Zoghman received the Servant Medal from the CNRS a prize given every two years with an amount of €10,000. Notable works In September 1979 Mebkhout presented the Riemann–Hilbert correspondence, which is a generalization of Hilbert's twenty-first problem to higher dimensions. The original setting was for Riemann surfaces, where it was about the existence of regular differential equations with prescribed monodromy groups. In higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension > 1. Certain systems of partial differential equations (linear and having very special properties for their soluti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involutive System
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of ''r'' vector fields mesh into coordinate grids on ''r''-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds. Contact geometry studies 1-forms that maximally violates the assumptions of Frobenius' theorem. An example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Algebra
In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. In the simplest case, these are differential operators. Let F be a field, and let F /math> be the ring of polynomials in one variable with coefficients in F. Then the corresponding Weyl algebra consists of differential operators of form : f_m(x) \partial_x^m + f_(x) \partial_x^ + \cdots + f_1(x) \partial_x + f_0(x) This is the first Weyl algebra A_1. The ''n''-th Weyl algebra A_n are constructed similarly. Alternatively, A_1 can be constructed as the quotient of the free algebra on two generators, ''q'' and ''p'', by the ideal generated by ( ,q- 1). Similarly, A_n is obtained by quotienting the free algebra on ''2n'' generators by the ideal generated by ( _i,q_j- \delta_), \quad \forall i, j = 1, \dots, nwhere \delta_ is the K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero. That is, is the smallest positive number such that: : \underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: : \underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of rngs (see '); for (unital) rings the two definitions are equivalent due to their distributive law. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
