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Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the five mathematicians to have won the Fields Medal, the Wolf Prize, and the Abel Prize (the others being Jean-Pierre Serre, Serre, John G. Thompson, Thompson, Pierre Deligne, Deligne, and Grigory Margulis, Margulis.) Early life and career Milnor was born on February 20, 1931, in Orange, New Jersey. His father was J. Willard Milnor and his mother was Emily Cox Milnor. As an undergraduate at Princeton University he was named a William Lowell Putnam Mathematical Competition, Putnam Fellow in 1949 and 1950 and also proved the Fáry–Milnor theorem when he was only 19 years old. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of Ralph Fox, Ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor K-theory
In mathematics, Milnor K-theory is an algebraic invariant (denoted K_*(F) for a field F) defined by as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms. Before Milnor K-theory was defined, there existed ad-hoc definitions for K_1 and K_2. Fortunately, it can be shown Milnor is a part of algebraic , which in general is the easiest part to compute. Definition Motivation After the definition of the Grothendieck group K(R) of a commutative ring, it was expected there should be an infinite set of invariants K_i(R) called higher groups, from the fact there exists a short exact sequence :K(R,I) \to K(R) \to K(R/I) \to 0 which should have a continuation by a long exact sequence. Note the group on the left is relative . This led to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exotic Sphere
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic"). The first exotic spheres were constructed by in dimension n = 7 as S^3- bundles over S^4. He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4. The classification of exotic spheres by showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum. Introduction The unit ''n''-sphere, S^n, is the set of all ('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor Number
In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If ''f'' is a complex-valued holomorphic function germ then the Milnor number of ''f'', denoted ''μ''(''f''), is either a nonnegative integer, or is infinite. It can be considered both a geometric invariant and an algebraic invariant. This is why it plays an important role in algebraic geometry and singularity theory. Algebraic definition Consider a holomorphic complex function germ : f : (\mathbb^n,0) \to (\mathbb,0) \ and denote by \mathcal_n the ring of all function germs (\mathbb^n,0) \to (\mathbb,0). Every level of a function is a complex hypersurface in \mathbb^n, therefore we will call f a hypersurface singularity. Assume it is an isolated singularity: in case of holomorphic mappings we say that a hypersurface singularity f is singular at 0 \in \mathbb^n if its gradient \nabla f is zero at 0 , a singular point is isolated if it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor Fibration
In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book ''Singular Points of Complex Hypersurfaces'' (Princeton University Press, 1968) and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly encountered in the mathematical literature. These were introduced to study isolated singularities by constructing numerical invariants related to the topology of a smooth deformation of the singular space. Definition Let f(z_0,\dots,z_n) be a non-constant polynomial function of n+1 complex variables z_0,\dots,z_n where the vanishing locus of :f(z)\ \text\ \frac(z) is only at the origin, meaning the associated variety X = V(f) is not smooth at the origin. Then, for K = X \cap S^_ (a sphere inside \mathbb^ of radius \varepsilon > 0) the Milnor fibrationpg 68 associated to f is defined as the map :\phi\colon (S_\varepsilon^\setminus K) \to S^1\ \text \ x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor Map
In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book ''Singular Points of Complex Hypersurfaces'' (Princeton University Press, 1968) and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly encountered in the mathematical literature. These were introduced to study isolated singularities by constructing numerical invariants related to the topology of a smooth deformation of the singular space. Definition Let f(z_0,\dots,z_n) be a non-constant polynomial function of n+1 complex variables z_0,\dots,z_n where the vanishing locus of :f(z)\ \text\ \frac(z) is only at the origin, meaning the associated variety X = V(f) is not smooth at the origin. Then, for K = X \cap S^_ (a sphere inside \mathbb^ of radius \varepsilon > 0) the Milnor fibrationpg 68 associated to f is defined as the map :\phi\colon (S_\varepsilon^\setminus K) \to S^1\ \text \ x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surgery Theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold ''M'' of dimension n=p+q+1, could be described as removing an imbedded sphere of dimension ''p'' from ''M''. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. More technically, the idea is to start with a well-understood manifold ''M'' and perform surgery on it to produce a manifold ''M''′ having some desired property, in such a way th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Švarc–Milnor Lemma
In the mathematical subject of geometric group theory, the Švarc–Milnor lemma (sometimes also called Milnor–Švarc lemma, with both variants also sometimes spelling Švarc as Schwarz) is a statement which says that a group G, equipped with a "nice" discrete action, discrete isometric Group action (mathematics), action on a metric space X, is Quasi-isometry, quasi-isometric to X. This result goes back, in different form, before the notion of quasi-isometry was formally introduced, to the work of Albert Schwarz, Albert S. Schwarz (1955) and John Milnor (1968). Pierre de la Harpe called the Švarc–Milnor lemma ``the ''fundamental observation in geometric group theory''"Pierre de la Harpe, ''Topics in geometric group theory'. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ; p. 87 because of its importance for the subject. Occasionally the name "fundamental observation in geometric group theory" is now used for this statement, instead of ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor–Wood Inequality
In mathematics, more specifically in differential geometry and geometric topology, the Milnor–Wood inequality is an obstruction to endow circle bundles over surfaces with a flat structure. It is named after John Milnor and John W. Wood. Flat bundles For linear bundles, flatness is defined as the vanishing of the curvature form of an associated connection. An arbitrary smooth (or topological) ''d''-dimensional fiber bundle is flat if it can be endowed with a foliation of codimension d that is transverse to the fibers. The inequality The Milnor–Wood inequality is named after two separate results that were proven by John Milnor and John W. Wood. Both of them deal with orientable circle bundles over a closed oriented surface \Sigma_g of positive genus ''g''. Theorem (Milnor, 1958) Let \pi\colon E \to \Sigma_g be a flat oriented linear circle bundle. Then the Euler number of the bundle satisfies , e(\pi), \leq g -1. Theorem (Wood, 1971) Let \pi\colon E \to \Sigma_g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plumbing (mathematics)
In the mathematical field of geometric topology, among the techniques known as surgery theory, the process of plumbing is a way to create new manifolds out of disk bundles. It was first described by John Milnor and subsequently used extensively in surgery theory to produce manifolds and normal maps with given surgery obstructions. Definition Let \xi_i=(E_i,M_i,p_i) be a rank ''n'' vector bundle over an ''n''-dimensional smooth manifold M_i for ''i'' = 1,2. Denote by D(E_i) the total space of the associated (closed) disk bundle D(\xi_i)and suppose that \xi_i, M_i and D(E_i)are oriented in a compatible way. If we pick two points x_i\in M_i, ''i'' = 1,2, and consider a ball neighbourhood of x_i in M_i, then we get neighbourhoods D^n_i\times D^n_i of the fibre over x_i in D(E_i). Let h:D^n_1\rightarrow D^n_2 and k:D^n_1\rightarrow D^n_2 be two diffeomorphisms (either both orientation preserving or reversing). The plumbing of D(E_1) and D(E_2) at x_1 and x_2 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fáry–Milnor Theorem
In the mathematical theory of knots, the Fáry–Milnor theorem, named after István Fáry and John Milnor, states that three-dimensional smooth curves with small total curvature must be unknotted. The theorem was proved independently by Fáry in 1949 and Milnor in 1950. It was later shown to follow from the existence of quadrisecants . Statement If ''K'' is any closed curve in Euclidean space that is sufficiently smooth to define the curvature κ at each of its points, and if the total absolute curvature is less than or equal to 4π, then ''K'' is an unknot, i.e.: : \text\ \oint_K , \kappa(s), \, \mathrms \le 4 \pi,\ \text\ K\ \text. The contrapositive tells us that if ''K'' is not an unknot, i.e. ''K'' is not isotopic to the circle, then the total curvature will be strictly greater than 4π. Notice that having the total curvature less than or equal to 4 is merely a sufficient condition for ''K'' to be an unknot; it is not a necessary condition In logic and mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rokhlin's Theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold ''M'' has a spin structure (or, equivalently, the second Stiefel–Whitney class w_2(M) vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group H^2(M), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952. Examples *The intersection form on ''M'' ::Q_M\colon H^2(M,\Z)\times H^2(M,\Z)\rightarrow \mathbb :is unimodular on \Z by Poincaré duality, and the vanishing of w_2(M) implies that the intersection form is even. By a theorem of Cahit Arf, any even unimodular lattice has signature divisible by 8, so Rokhlin's theorem forces one extra factor of 2 to divide the signature. *A K3 surface is compact, 4 dimensional, and w_2(M) vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem. *A complex surface in \mathbb^3 of degree d is spin if and only if d is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor–Thurston Kneading Theory
The Milnor–Thurston kneading theory is a mathematical theory which analyzes the iterates of piecewise monotone mappings of an interval into itself. The emphasis is on understanding the properties of the mapping that are invariant under topological conjugacy. The theory had been developed by John Milnor and William Thurston in two widely circulated and influential Princeton preprints from 1977 that were revised in 1981 and finally published in 1988. Applications of the theory include piecewise linear models, counting of fixed points, computing the total variation, and constructing an invariant measure with maximal entropy. Short description Kneading theory provides an effective calculus for describing the qualitative behavior of the iterates of a piecewise monotone mapping ''f'' of a closed interval ''I'' of the real line into itself. Some quantitative invariants of this discrete dynamical system, such as the ''lap numbers'' of the iterates and the Artin–Mazur zeta functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
