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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. The 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 sym ... [...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 o ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Bernstein
Joseph Bernstein (sometimes spelled I. N. Bernshtein; he, יוס(י)ף נאומוביץ ברנשטיין; russian: Иосиф Наумович Бернштейн; 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 anti-semitism 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 Academ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zoghman Mebkhout
Zoghman Mebkhout (born 1949 ) (مبخوت زغمان) is a French-Algerian mathematician. He is known for his work in algebraic analysis, geometry and representation theory, more precisely on the theory of ''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 solutions) and possib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Masaki Kashiwara
is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Kashiwara made leading contributions towards algebraic analysis, microlocal analysis, D-module, ''D''-module theory, Hodge theory, sheaf theory and representation theory. Kashiwara and Sato established the foundations of the theory of systems of linear partial differential equations with analytic coefficients, introducing a cohomological approach that follows the spirit of Grothendieck's theory of scheme (mathematics), schemes. Joseph Bernstein, Bernstein introduced a similar approach in the polynomial coefficients case. Kashiwara's master thesis states the foundations of D-module, ''D''-module theory. His PhD thesis proves the rationality of the roots of b-functions (Bernstein–Sato polynomials), using ''D''-module theory and resolution of singularities. He was a plenary speaker at International Congress of Mathematicians, 1978, Helsinki and an invited speaker, 1990, Kyoto. He is a member o ... [...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 equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotangent Bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories. Formal Definition Let ''M'' be a smooth manifold and let ''M''×''M'' be the Cartesian product of ''M'' with itself. The diagonal mapping Δ sends a point ''p'' in ''M'' to the point (''p'',''p'') of ''M''×''M''. The image of Δ is called the diagonal. Let \mathcal be the sheaf of germs of smooth functions on ''M''×''M'' which vanish on the diagona ... [...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. Introduction In its most elementary form, the theorem addresses the problem of finding a maximal set of ind ... [...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 made remarkab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Variety
In mathematical analysis, the characteristic variety of a microdifferential operator ''P'' is an algebraic variety that is the zero set of the principal symbol of ''P'' in the cotangent bundle. It is invariant under a quantized contact transformation. The notion is also defined more generally in commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent .... A basic theorem says a characteristic variety is involutive. References * M. Sato, T. Kawai, and M. Kashiwara: Microfunctions and Pseudo-differential Equations. Lecture note in Math., No. 287, Springer, Berlin-Heidelberg-New York, pp. 265–529 (1973) {{analysis-stub Algebraic varieties Mathematical analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More precisely, let ''F'' be the underlying field, and let ''F'' 'X''be the ring of polynomials in one variable, ''X'', with coefficients in ''F''. Then each ''fi'' lies in ''F'' 'X'' ''∂X'' is the derivative with respect to ''X''. The algebra is generated by ''X'' and ''∂X''. The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension. The Weyl algebra is isomorphic to the quotient of the free algebra on two generators, ''X'' and ''Y'', by the ideal generated by the element :YX - XY = 1~. The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The ''n''-th Weyl algebra, ''An'', is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
