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Norman Steenrod
Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of Michigan (A.B. 1932). After receiving a master's degree from Harvard University in 1934, he enrolled at Princeton University. He completed his Ph.D. under the direction of Solomon Lefschetz, with a thesis titled ''Universal homology groups''. Steenrod held positions at the University of Chicago from 1939 to 1942, and the University of Michigan from 1942 to 1947. He moved to Princeton University in 1947, and remained on the Faculty there for the rest of his career. He was editor of the Annals of Mathematics and a member of the National Academy of Sciences. He died in Princeton, survived by his wife, the former Carolyn Witter, and two children. Work Thanks to Lefschetz and others, the cup product structure of cohomology was understood by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dayton, Ohio
Dayton () is a city in Montgomery County, Ohio, United States, and its county seat. It is the List of cities in Ohio, sixth-most populous city in Ohio, with a population of 137,644 at the 2020 United States census, 2020 census. The Dayton metropolitan area had 814,049 residents and is the state's fourth-largest metropolitan area. Dayton is located within Ohio's Miami Valley region, north of Cincinnati and west-southwest of Columbus, Ohio, Columbus. Dayton was founded in 1796 along the Great Miami River and named after Jonathan Dayton, a Founding Fathers of the United States, Founding Father who owned a significant amount of land in the area. It grew in the 19th century as a canal town and was home to many patents and inventors, most notably the Wright brothers, who developed the first successful motor-operated airplane. It later developed an industrialized economy and was home to the Dayton Project, a branch of the larger Manhattan Project, to develop polonium triggers used in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George W
George Walker Bush (born July 6, 1946) is an American politician and businessman who was the 43rd president of the United States from 2001 to 2009. A member of the Bush family and the Republican Party (United States), Republican Party, he is the eldest son of the 41st president, George H. W. Bush, and was the 46th governor of Texas from 1995 to 2000. Bush flew warplanes in the Texas Air National Guard in his twenties. After graduating from Harvard Business School in 1975, he worked in the oil industry. He later co-owned the Major League Baseball team Texas Rangers (baseball), Texas Rangers before being elected governor of Texas 1994 Texas gubernatorial election, in 1994. Governorship of George W. Bush, As governor, Bush successfully sponsored legislation for tort reform, increased education funding, set higher standards for schools, and reformed the criminal justice system. He also helped make Texas the Wind power in Texas, leading producer of wind-generated electricity in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miami University
Miami University (informally Miami of Ohio or simply Miami) is a public university, public research university in Oxford, Ohio, United States. Founded in 1809, it is the second-oldest List of colleges and universities in Ohio, university in Ohio and the tenth-oldest public university in the United States. The university enrolls 18,600 students in Oxford and maintains Satellite campus, regional campuses in nearby Miami University Hamilton, Hamilton, Miami University Middletown, Middletown, and Miami University Voice of America Learning Center, West Chester. Miami also operates the international Miami University Dolibois European Center, Dolibois European Center in Differdange, Luxembourg. Miami University provides a liberal arts education; it offers more than 120 undergraduate degree programs and over 70 graduate degree programs within its seven schools and colleges in architecture, business, engineering, humanities and the sciences. It is a member of the University System of Ohi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colloquium Lectures (AMS)
The Colloquium Lecture of the American Mathematical Society is a special annual session of lectures. History The origins of the Colloquium Lectures date back to the 1893 International Congress of Mathematics, held in connection with the Chicago World's Fair (1893), Chicago World's Fair, where the German mathematician Felix Klein gave the opening address. After the Congress, Klein was invited by one of its organiser, his former student Henry Seely White, to deliver a two-week-long series of lectures at Northwestern University in Evanston, Illinois, Evanston. In February 1896, White proposed in a letter to Thomas Fiske to repeat the experience of the Evanston lectures, by organising a series of longer talks "for increasing the utility of the American Mathematical Society". The two of them, together with E. H. Moore, William Fogg Osgood, William Osgood, Frank Nelson Cole, Frank Cole, Alexander Ziwet, and Frank Morley, wrote later an open letter to the AMS, asking the society to sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Guggenheim Fellowships Awarded In 1950
One hundred and fifty-eight Guggenheim Fellowships were awarded in 1950. This marked the 25th anniversary of the fellowship. 1950 U.S. and Canadian Fellows 1950 Latin American and Caribbean Fellows See also * Guggenheim Fellowship * List of Guggenheim Fellowships awarded in 1949 * List of Guggenheim Fellowships awarded in 1951 References {{DEFAULTSORT:List Of Guggenheim Fellowships Awarded In 1950 1950 Events January * January 1 – The International Police Association (IPA) – the largest police organization in the world – is formed. * January 5 – 1950 Sverdlovsk plane crash, Sverdlovsk plane crash: ''Aeroflot'' Lisunov Li-2 ... Guggenheim Guggenheim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local System
In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between homology theory, cohomology with coefficients in a fixed abelian group ''A'', and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. Local systems are the building blocks of more general tools, such as constructible sheaf, constructible and perverse sheaf, perverse sheaves. Definition Let ''X'' be a topological space. A local system (of abelian groups/Module over a ring, modules...) on ''X'' is a locally constant sheaf (of abelian groups/sheaf of modules, of modules...) on ''X''. In other words, a sheaf \mathcal is a local system if every point has an open neighborhood U such that the restricted sheaf \mathcal, _U is isomorphic to the sheafification of some constant presheaf. Equivalent definitions Path-connected spaces If ''X'' is C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Myers–Steenrod Theorem
Two theorems in the mathematical field of Riemannian geometry bear the name Myers–Steenrod theorem, both from a 1939 paper by Myers and Steenrod. The first states that every distance-preserving surjective map (that is, an isometry of metric spaces) between two connected Riemannian manifolds is a smooth isometry of Riemannian manifolds. A simpler proof was subsequently given by Richard Palais in 1957. The main difficulty lies in showing that a distance-preserving map, which is a priori only continuous, is actually differentiable. The second theorem, which is harder to prove, states that the isometry group \mathrm(M) of a connected \mathcal^2 Riemannian manifold M is a Lie group in a way that is compatible with the compact-open topology and such that the action \mathrm(M)\times M \longrightarrow M is \mathcal^1differentiable (in both variables). This is a generalization of the easier, similar statement when M is a Riemannian symmetric space: for instance, the group of isometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–Steenrod Axioms
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod. One can define a homology theory as a sequence of functors satisfying the Eilenberg–Steenrod axioms. The axiomatic approach, which was developed in 1945, allows one to prove results, such as the Mayer–Vietoris sequence, that are common to all homology theories satisfying the axioms. If one omits the dimension axiom (described below), then the remaining axioms define what is called an extraordinary homology theory. Extraordinary cohomology theories first arose in K-theory and cobordism. Formal definition The Eilenberg–Steenrod axioms apply to a sequence of functors H_n from the category of pairs (X,A) of topological spaces to the category of abelian groups, together with a n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steenrod Square
Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of Michigan (A.B. 1932). After receiving a master's degree from Harvard University in 1934, he enrolled at Princeton University. He completed his Ph.D. under the direction of Solomon Lefschetz, with a thesis titled ''Universal homology groups''. Steenrod held positions at the University of Chicago from 1939 to 1942, and the University of Michigan from 1942 to 1947. He moved to Princeton University in 1947, and remained on the Faculty there for the rest of his career. He was editor of the Annals of Mathematics and a member of the National Academy of Sciences. He died in Princeton, survived by his wife, the former Carolyn Witter, and two children. Work Thanks to Lefschetz and others, the cup product structure of cohomology was understood by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steenrod Problem
In mathematics, and particularly homology theory, Steenrod's Problem (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds. Formulation Let M be a closed, oriented manifold of dimension n, and let \in H_n(M) be its orientation class. Here H_n(M) denotes the integral, n-dimensional homology group of M. Any continuous map f\colon M\to X defines an induced homomorphism f_*\colon H_n(M)\to H_n(X). A homology class of H_n(X) is called realisable if it is of the form f_* /math> where \in H_n(M). The Steenrod problem is concerned with describing the realisable homology classes of H_n(X). Results All elements of H_k(X) are realisable by smooth manifolds provided k\le 6. Moreover, any cycle can be realized by the mapping of a pseudo-manifold. The assumption that ''M'' be orientable can be relaxed. In the case of non-orientable manifolds, every homology class of H_n(X,\Z_2), where \Z_2 denotes the integers modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
