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Donaldson's Theorem
In mathematics, and especially differential topology and gauge theory (mathematics), gauge theory, Donaldson's theorem states that a definite quadratic form, definite intersection form (4-manifold), intersection form of a Compact space, compact, orientability, oriented, smooth manifold of dimension 4 is diagonalizable matrix, diagonalizable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the . The original version of the theorem required the manifold to be simply connected, but it was later improved to apply to 4-manifolds with any fundamental group. History The theorem was proved by Simon Donaldson. This was a contribution cited for his Fields medal in 1986. Idea of proof Donaldson's proof utilizes the moduli space \mathcal_P of solutions to the Yang–Mills_equations#Anti-self-duality_equations, anti-self-duality equations on a principal bundle, principal \operatorname(2)-bundle P over the 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] |
Principal Bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with # An action of G on P, analogous to (x, g)h = (x, gh) for a product space (where (x, g) is an element of P and h is the group element from G; the group action is conventionally a right action). # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unless it is the product space X \times G, a principal bundle lacks a preferred choice of identity cross-section; it has no preferred analog of x \mapsto (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X \times G \to G that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre Classification Theorem
Serre may refer to: * Serre (surname) * Serre (grape), a red Italian wine grape * Serre (river), a tributary of the Oise in France * Serre, Campania, a town and comune in Salerno, Campania, Italy * Serre-lès-Puisieux, a village in Pas-de-Calais department, northern France * Serre Chevalier, a French ski resort in the Alps * Serre Calabresi, a mountain and hill area of Calabria, Italy See also * Serr, a surname * Serres (other) * La Serre (other) {{Disambiguation, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are important in physics because in general relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. Topological 4-manifolds The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of implies that the homeomorphism type of the manifold only depends on this intersection form, and on a \Z/2\Z invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and Kirb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unimodular Symmetric Bilinear Form
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1. The ''E''8 lattice and the Leech lattice are two famous examples. Definitions * A lattice is a free abelian group of finite rank with a symmetric bilinear form (·, ·). * The lattice is integral if (·,·) takes integer values. * The dimension of a lattice is the same as its rank (as a Z- module). * The norm of a lattice element ''a'' is (''a'', ''a''). * A lattice is positive definite if the norm of all nonzero elements is positive. * The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (''ai'', ''aj''), where the elements ''ai'' form a basis for the lattice. * An integral lattice is unimodular if its determinant is 1 or −1. * A unimodular lattice is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Freedman
Michael Hartley Freedman (born April 21, 1951) is an American mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional generalized Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists. Life and career Freedman was born in Los Angeles, California, in the United States. His father, Benedict Freedman, was an American Jewish aeronautical engineer, musician, writer, and mathematician. His mother, Nancy Mars Freedman, performed as an actress and also trained as an artist. His parents cowrote a series of novels together.. He entered the University of California, Berkeley, but dropped out after two semesters. In the same year he wrote a letter to Ralph Fox, a Princeton University professor at the time, and was admitted to the university's graduate school, where in 1968 he continued his studies and received a Ph.D. in 1973 for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#French, bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their disjoint union is the ''boundary'' of a compact manifold one dimension higher. The boundary of an (n+1)-dimensional manifold W is an n-dimensional manifold \partial W that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for Piecewise linear manifold, piecewise linear and topological manifolds. A ''cobordism'' between manifolds M and N is a compact manifold W whose boundary is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Projective Plane
In mathematics, the complex projective plane, usually denoted or is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \C^3, \qquad (Z_1,Z_2,Z_3)\neq (0,0,0) where, however, the triples differing by an overall rescaling are identified: :(Z_1,Z_2,Z_3) \equiv (\lambda Z_1,\lambda Z_2, \lambda Z_3); \quad \lambda \in \C, \qquad \lambda \neq 0. That is, these are homogeneous coordinates in the traditional sense of projective geometry. Topology The Betti numbers of the complex projective plane are :1, 0, 1, 0, 1, 0, 0, ..... The middle dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere, lying in the plane. The nontrivial homotopy groups of the complex projective plane are \pi_2=\pi_5=\mathbb. The fundamental group is trivial and all other higher homotopy groups are those of the 5-sphere, i.e. torsion. Algebraic geometry In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone (linear Algebra)
In linear algebra, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real vector space that is closed under positive scalar multiplication; that is, C is a cone if x\in C implies sx\in C for every . This is a broad generalization of the standard cone in Euclidean space. A convex cone is a cone that is also closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets. The definition of a convex cone makes sense in a vector space over any ordered field, although the field of real numbers is used most often. Definition A subset C of a vector space is a cone if x\in C implies sx\in C for every s>0. Here s>0 refers to (strict) positivity in the scalar field. Competing definitions Some other authors require ,\infty)C\subset C or even 0\in C. Some require a cone to be convex and/or satisfy C\cap-C\subset\. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karen Uhlenbeck
Karen Keskulla Uhlenbeck ForMemRS (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richardson Foundation Regents Chair. She is currently a distinguished visiting professor at the Institute for Advanced Study and a visiting senior research scholar at Princeton University. Uhlenbeck was elected to the American Philosophical Society in 2007. She won the 2019 Abel Prize for "her pioneering achievements in geometric partial differential equations, gauge theory, and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics." She is the first, and so far only, woman to win the prize since its inception in 2003. She donated half of the prize money to organizations which promote more engagement of women in research mathematics. Life and career Uhlenbeck was born in Clevel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Taubes
Clifford Henry Taubes (born February 21, 1954) is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology. His brother is the journalist Gary Taubes. Early career Taubes received his B.A. from Cornell University in 1975 and his Ph.D. in physics in 1980 from Harvard University under the direction of Arthur Jaffe, having proven results collected in about the existence of solutions to the Landau–Ginzburg vortex equations and the Bogomol'nyi monopole equations. Soon, he began applying his gauge-theoretic expertise to pure mathematics. His work on the boundary of the moduli space of solutions to the Yang-Mills equations was used by Simon Donaldson in his proof of Donaldson's theorem on diagonizability of intersection forms. He proved in that R4 has an uncountable number of smooth structures (see also exotic R4), and (with Raoul Bott in ) proved Witten's rigidity theorem on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donaldson's Theorem Cobordism
Donaldson's, previously known as the L. S. Donaldson Company, headquartered in Minneapolis, Minnesota, is a defunct department store company. History Scottish immigrant William Donaldson opened a small store in Minneapolis in 1881, located at 310 Nicollet Avenue. In 1883, William and his brother Lawrence purchased a -story store named Colton and Company, featuring a large expanse of glass block. The Donaldson brothers department store was known in its early years as "Donaldson's Glass Block Store" because of this distinctive design feature. In 1888, the original building was demolished, and replaced with a five-story building featuring a dome on top, elevators, and rows of plate glass windows. By 1899, William had died, and Lawrence renamed the company the "L.S. Donaldson Company". The store continued to expand, which culminated in the construction of a new $2,000,000 (~$ in ) eight-story building, taking up an entire block of Nicollet from Sixth Street to Seventh Street, topped ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
