|
Grothendieck–Katz P-curvature Conjecture
In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density theorem considered as the polynomial case. It is a conjecture of Alexander Grothendieck from the late 1960s, and apparently not published by him in any form. The general case remains unsolved, despite recent progress; it has been linked to geometric investigations involving algebraic foliations. Formulation In a simplest possible statement the conjecture can be stated in its essentials for a vector system written as :dv/dz = A(z)v for a vector ''v'' of size ''n'', and an ''n''-by-''n'' matrix ''A'' of algebraic functions with algebraic number coefficients. The question is to give a criterion for when there is a ''full set'' of algebraic function solutions, meaning a fundamental matrix (i.e. ''n'' vector solutions put into a block matrix). Fo ... [...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] |
Nicholas Katz
Nicholas Michael Katz (; born December 7, 1943) is an American mathematician, working in arithmetic geometry, particularly on ''p''-adic methods, monodromy and moduli problems, and number theory. He is currently a professor of Mathematics at Princeton University and an editor of the journal ''Annals of Mathematics''. Life and work Katz graduated from Johns Hopkins University (BA 1964) and from Princeton University, where in 1965 he received his master's degree and in 1966 he received his doctorate under supervision of Bernard Dwork with thesis ''On the Differential Equations Satisfied by Period Matrices''. After that, at Princeton, he was an instructor, an assistant professor in 1968, associate professor in 1971 and professor in 1974. From 2002 to 2005 he was the chairman of faculty there. He was also a visiting scholar at the University of Minnesota, the University of Kyoto, Paris VI, Orsay Faculty of Sciences, the Institute for Advanced Study and the IHES. While in F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory. History One of the origins of the mathematical theory of arithmetic groups is algebraic number theory. The classical reduction theory of quadratic and Hermitian forms by Charles Hermite, Hermann Minkowski and others can be seen as computing fundamental domains for the action of certain arithmetic groups on the relevant symmetric spaces. The topic was related to Minkowski's geometry of numbers and the early development of the study of arithmetic invariant of number fields such as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superrigidity
In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group ''G'' can, under some circumstances, be as good as a representation of ''G'' itself. That this phenomenon happens for certain broadly defined classes of lattices inside semisimple groups was the discovery of Grigory Margulis, who proved some fundamental results in this direction. There is more than one result that goes by the name of ''Margulis superrigidity''. One simplified statement is this: take ''G'' to be a simply connected semisimple real algebraic group in ''GL''''n'', such that the Lie group of its real points has real rank at least 2 and no compact factors. Suppose Γ is an irreducible lattice in G. For a local field ''F'' and ρ a linear representation of the lattice Γ of the Lie group, into ''GL''''n'' (''F''), assume the image ρ(Γ) is not relatively compact (in the topology arising from ''F'') a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Manin Connection
In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space ''S'' of a family of algebraic varieties V_s. The fibers of the vector bundle are the de Rham cohomology groups H^k_(V_s) of the fibers V_s of the family. It was introduced by for curves ''S'' and by in higher dimensions. Flat sections of the bundle are described by differential equations; the best-known of these is the Picard–Fuchs equation, which arises when the family of varieties is taken to be the family of elliptic curves. In intuitive terms, when the family is locally trivial, cohomology classes can be moved from one fiber in the family to nearby fibers, providing the 'flat section' concept in purely topological terms. The existence of the connection is to be inferred from the flat sections. Intuition Consider a smooth morphism of schemes X\to B over characteristic 0. If we consider these spaces as complex analytic spaces, then the Ehresmann fibration theorem te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard–Fuchs Equation
In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves. Definition Let :j=\frac be the j-invariant with g_2 and g_3 the Elliptic modular function, modular invariants of the elliptic curve in Weierstrass equation, Weierstrass form: :y^2=4x^3-g_2x-g_3.\, Note that the ''j''-invariant is an isomorphism from the Riemann surface \mathbb/\Gamma to the Riemann sphere \mathbb\cup\; where \mathbb is the upper half-plane and \Gamma is the modular group. The Picard–Fuchs equation is then :\frac + \frac \frac + \frac y=0.\, Written in Schwarzian derivative, Q-form, one has :\frac + \frac f=0.\, Solutions This equation can be cast into the form of the hypergeometric differential equation. It has two linearly independent solutions, called the periods of elliptic functions. The ratio of the two periods is equal to the half-period ratio, period ratio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Symmetric Variety
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (''M'', ''g'') is said to be symmetric if and only if, for each point ''p'' of ''M'', there exists an isometry of ''M'' fixing ''p'' and acting on the tangent space T_pM as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark Kisin
Mark Kisin is a mathematician known for work in algebraic number theory and arithmetic geometry. In particular, he is known for his contributions to the study of p-adic representations and p-adic cohomology. Born in Vilnius, Lithuania and raised from the age of five in Melbourne, Australia, he won a silver medal at the International Mathematical Olympiad in 1989 and received his B.Sc. from Monash University in 1991. He received his Ph.D. from Princeton University in 1998 under the direction of Nick Katz. From 1998 to 2001 he was a Research Fellow at the University of Sydney, after which he spent three years at the University of Münster. After six years at the University of Chicago, Kisin took the post in 2009 of professor of mathematics at Harvard University. He was elected a Fellow of the Royal Society in 2008. He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Number Theory". In 2012 he became a fellow of the American Mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benson Farb
Benson Stanley Farb (born October 25, 1967) is an American mathematician at the University of Chicago. His research fields include geometric group theory and low-dimensional topology. Early life A native of Norristown, Pennsylvania, Farb earned his bachelor's degree from Cornell University. In 1994, he obtained his doctorate from Princeton University, under supervision of William Thurston. Career Farb has advised over 40 students, including Pallavi Dani, Kathryn Mann, Dan Margalit, Karin Melnick and Andrew Putman. In 2012 Farb became a fellow of the American Mathematical Society. In 2014 he was an invited speaker at the International Congress of Mathematicians in Seoul, speaking in the section on Topology. He was elected to the American Academy of Arts and Sciences in 2021. In 2024 he was awarded the Leroy P. Steele Prize for Mathematical Exposition. Books *Reviews of ''A Primer on Mapping Class Groups'' * * * * Personal life Farb married Amie Wilkinson, professor of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical evidence, empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Closure
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups, including orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
