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Donaldson's Theorem
In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalisable. 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 anti-self-duality equations on a principal \operatorname(2)-bundle P over the four-manifold X. By the Atiyah–Singer index theorem, the dimension of the moduli space is given by :\dim \mathcal = 8k - 3(1-b_1(X) + b_+(X)), where c_2(P)=k, b_1(X) is the first Betti number of X and b ... [...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] |
Atiyah–Singer Index Theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics. History The index problem for elliptic differential operators was posed by Israel Gel'fand. He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Friedrich Hirzebruch and Armand Borel had proved the integrality of the  genus of a spin manifold, and At ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Structure
In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If ''M'' is already a topological manifold, it is required that the new topology be identical to the existing one. Definition For a natural number ''n'' and some ''k'' which may be a non-negative integer or infinity, an ''n''-dimensional ''C''''k'' differential structure is defined using a ''C''''k''-atlas, which is a set of bijections called charts between a collection of subsets of ''M'' (whose union is the whole of ''M''), and a set of open subsets of \mathbb^: :\varphi_:M\supset W_\rightarrow U_\subset\mathbb^ which are ''C''''k''-compatible (in the sense defined below): Each such map provides a way in which certain subsets of the manifold may be viewed as being like open subsets of \mathbb^ but th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Manifold
In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable manifolds are topological manifolds equipped with a differential structure). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure. Formal definition A topological space ''X'' is called locally Euclidean if there is a non-negative integer ''n'' such that every point in ''X'' has a neighborhood which is homeomorphi ... [...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 Serre Chevalier () is a major ski resort in Southeastern France, near the Italian border, located in the Hautes-Alpes department, Provence-Alpes-Côte d'Azur region. Situated to the northeast of Écrins National Park in the French Alps, the re ..., 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 In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which t ... 4-manifold. Topological 4-manifolds The homotopy type of a simply connected compact 4-manifold only depends on the intersection form (4-manifold), intersection ... [...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 professor at the time, and was admitted to graduate school so in 1968 he continued his studies at Princeton University where he received Ph.D. degree in 1973 fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (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 ∂''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 and topological manifolds. A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'', \partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Projective Plane
In mathematics, the complex projective plane, usually denoted P2(C), 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 \mathbf^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 \mathbf,\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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone (linear Algebra)
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closure (mathematics), closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . When the scalars are real numbers, or belong to an ordered field, one generally calls a cone a subset of a vector space that is closed under multiplication by a ''positive scalar''. In this context, a convex cone is a cone that is 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. In this article, only the case of scalars in an ordered field is considered. Definition A subset ''C'' of a vector space ''V'' over an ordered field ''F'' is a cone (or sometimes called a linear cone) if for each ''x'' in ''C'' and positive scalar ''α'' in ''F'', the product ''αx'' is in ''C''. Note that s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karen Uhlenbeck
Karen Keskulla Uhlenbeck (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 Cleveland, Ohio, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
