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List Of Things Named After John Milnor
{{Short description, none Things named after an American mathematician John Milnor: * Barratt–Milnor sphere * Fáry–Milnor theorem * Milnor conjecture (K-theory), Milnor conjecture in algebraic K-theory * Milnor conjecture (knot theory), Milnor conjecture in knot theory * Milnor conjecture (Ricci), Milnor conjecture concerning manifolds with nonnegative Ricci curvature * Exotic_sphere#Milnor's_construction, Milnor construction * Milnor K-theory * Milnor fibration * Brunnian_link#Classification, Milnor invariants * Plumbing_(mathematics)#Milnor_manifold, Milnor manifold * Milnor map * Milnor–Moore theorem * Milnor number * Milnor ring * Milnor sphere * Milnor theorem * Milnor–Thurston kneading theory * Milnor Frame concerning left invariant metrics on three-dimensional Lie groups * Milnor–Wood inequality * Švarc–Milnor lemma Lists of things named after mathematicians, Milnor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John 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 Serre, Thompson, Deligne, and 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 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 Robert H. Fox. He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after completi ... [...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] |
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] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor Frame
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 Serre, Thompson, Deligne, and 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 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 Robert H. Fox. He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after comp ... [...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] |
Milnor Theorem
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 Serre, Thompson, Deligne, and 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 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 Robert H. Fox. He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor 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 Ring
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 ... [...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–Moore Theorem
In algebra, the Milnor–Moore theorem, introduced by classifies an important class of Hopf algebras, of the sort that often show up as cohomology rings in algebraic topology. The theorem states: given a connected, graded, cocommutative Hopf algebra ''A'' over a field of characteristic zero with \dim A_n . In algebraic topology, the term usually refers to the corollary of the aforementioned result, that for a pointed, simply connected space ''X'', the following isomorphism holds: :U(\pi_(\Omega X) \otimes \Q) \cong H_(\Omega X;\Q), where \Omega X denotes the loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolo ... of ''X'', compare with Theorem 21.5 from . This work may also be compared with that of . References * * * * * * External links * Theorems abou ... [...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] |
