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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] |
Beno Eckmann
Beno Eckmann (31 March 1917 – 25 November 2008) was a Swiss mathematician who made contributions to algebraic topology, homological algebra, group theory, and differential geometry. Life Born in Bern, Eckmann received his master's degree from Eidgenössische Technische Hochschule Zürich (ETH) in 1939. Later he studied there under Heinz Hopf, obtaining his Ph.D. in 1941. Eckmann was the 2008 recipient of the Albert Einstein Medal. Legacy Calabi–Eckmann manifolds, Eckmann–Hilton duality, the Eckmann–Hilton argument, and the Eckmann–Shapiro lemma are named after Eckmann. Family Eckmann's son is mathematical physicist Mathematical physics refers to the development of mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developmen ... Jean-Pierre Eckmann. [...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] |
Sloan Fellowship
The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This program is one of the oldest of its kind in the United States. Fellowships were initially awarded in physics, chemistry, and mathematics. Awards were later added in neuroscience (1972), economics (1980), computer science (1993), computational and evolutionary molecular biology (2002), and ocean sciences or earth systems sciences (2012). Winners of these two-year fellowships are awarded $75,000, which may be spent on any expense supporting their research. From 2012 through 2020, the foundation awarded 126 research fellowship each year; in 2021, 128 were awarded, and 118 were awarded in 2022. Eligibility and selection To be eligible, a candidate must hold a Ph.D. or equivalent degree and must be a member of the faculty of a college, university, or other degree-granting institution in the United Sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Putnam Fellow
The William Lowell Putnam Mathematical Competition, often abbreviated to Putnam Competition, is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the United States and Canada (regardless of the students' nationalities). It awards a scholarship and cash prizes ranging from $250 to $2,500 for the top students and $5,000 to $25,000 for the top schools, plus one of the top five individual scorers (designated as ''Putnam Fellows'') is awarded a scholarship of up to $12,000 plus tuition at Harvard University (Putnam Fellow Prize Fellowship), the top 100 individual scorers have their names mentioned in the American Mathematical Monthly (alphabetically ordered within rank), and the names and addresses of the top 500 contestants are mailed to all participating institutions. It is widely considered to be the most prestigious university-level mathematics competition in the world, and its difficulty is such that the median score i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dusa McDuff
Dusa McDuff FRS CorrFRSE (born 18 October 1945) is an English mathematician who works on symplectic geometry. She was the first recipient of the Ruth Lyttle Satter Prize in Mathematics, was a Noether Lecturer, and is a Fellow of the Royal Society. She is currently the Helen Lyttle Kimmel '42 Professor of Mathematics at Barnard College. Personal life and education Margaret Dusa Waddington was born in London, England, on 18 October 1945 to Edinburgh architect Margaret Justin Blanco White, second wife of biologist Conrad Hal Waddington, her father. Her sister is the anthropologist Caroline Humphrey, and she has an elder half-brother C. Jake Waddington by her father's first marriage. Her mother was the daughter of Amber Reeves, the noted feminist, author and lover of H. G. Wells. McDuff grew up in Scotland where her father was Professor of Genetics at the University of Edinburgh. McDuff was educated at St George's School for Girls in Edinburgh and, although the standard was lowe ... [...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] |
Spectral Geometry
Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems. Inverse problems seek to identify features of the geometry from information about the eigenvalues of the Laplacian. One of the earliest results of this kind was due to Hermann Weyl who used David Hilbert's theory of integral equation in 1911 to show that the volume of a bounded domain in Euclidean space can be determined from the asymptotic behavior of the eigenvalues for the Dirichlet boundary value problem of the Laplace operator. This question is usually expressed as " Can one hear the shape of a drum?", the popula ... [...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] |
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] |
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] |
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] |
