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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] |
Steenrod Square
In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology. For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, consisting of all stable cohomology operations for mod p cohomology. It is generated by the Steenrod squares introduced by for p=2, and by the Steenrod reduced pth powers introduced in and the Bockstein homomorphism for p>2. The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory. Cohomology operations A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring R, the cup product squaring operation yields a family of cohomology operations: :H^n(X;R) \to H^(X;R) :x \mapsto x \smile x. Cohomology operations need not be homomorphisms of graded rings; see the Cartan formula below. Th ... [...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.http://www.math.uiuc.edu/K-theory/0245/survey.pdf 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a Jewish family. He spent much of his career as a professor at Columbia University. He earned his Ph.D. from University of Warsaw in 1936, with thesis ''On the Topological Applications of Maps onto a Circle''; his thesis advisors were Kazimierz Kuratowski and Karol Borsuk. He died in New York City in January 1998. Career Eilenberg's main body of work was in algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (and the Eilenberg–Steenrod axioms are named for the pair), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory. Eilenberg was a member of Bourbaki and, with Henri Cartan, wrote the 1956 book ''Homological Algebra''. La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
José Adem
José Adem (27 October 1921 – 14 February 1991) was a Mexican mathematician who worked in algebraic topology, and proved the Adem relations between Steenrod squares. Life and education Born José Adem Chahín in Tuxpan, Veracruz, (published his works as José Adem), Adem showed an interest in mathematics from an early age, and moved to Mexico City in 1941 to pursue a degree in engineering and mathematics. He obtained his B.S. in mathematics from the National Autonomous University of Mexico (UNAM) in 1949. During this time met Solomon Lefschetz, a famous algebraic topologist who was spending prolonged periods of time in Mexico. Lefschetz recognized Adem's mathematical talent, and sent him as a doctoral student to Princeton University where he graduated in 1952. His dissertation, ''Iterations of the squaring operations in algebraic topology'', was written under the supervision of Norman Steenrod and introduced what are now called the Adem relations. His brother is geophysicist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Gitler Hammer
Samuel Carlos Gitler Hammer (July 14, 1933 – September 9, 2014) CINVESTAV, retrieved 2012-05-19.Member biography , Colegio Nacional, retrieved 2012-05-19. was a Mexican mathematician. He was an expert in and is known for the |
Edwin Spanier
Edwin Henry Spanier (August 8, 1921 – October 11, 1996) was an American mathematician at the University of California at Berkeley, working in algebraic topology. He co-invented Spanier–Whitehead duality and Alexander–Spanier cohomology, and wrote what was for a long time the standard textbook on algebraic topology . Spanier attended the University of Minnesota, graduating in 1941. During World War II, he served in the United States Army Signal Corps. He received his Ph.D. degree from the University of Michigan in 1947 for the thesis ''Cohomology Theory for General Spaces'' written under the direction of Norman Steenrod. After spending a year as a research fellow at the Institute for Advanced Study in Princeton, New Jersey, in 1948 he was appointed to the faculty of the University of Chicago, and then a professor at UC Berkeley in 1959. He had 17 doctoral students, including Morris Hirsch and Elon Lages Lima. Publications * References * Retrieved on 2008-01-17 * Re ... [...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 invariants that classify topological spaces up to homeomorphism, though usually most classify up to 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 of algebraic topology 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 about the basic shape, or holes, of a topological space. Homolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solomon Lefschetz
Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. Life He was born in Moscow, the son of Alexander Lefschetz and his wife Sarah or Vera Lifschitz, Jewish traders who used to travel around Europe and the Middle East (they held Ottoman passports). Shortly thereafter, the family moved to Paris. He was educated there in engineering at the École Centrale Paris, but emigrated to the US in 1905. He was badly injured in an industrial accident in 1907, losing both hands. He moved towards mathematics, receiving a Ph.D. in algebraic geometry from Clark University in Worcester, Massachusetts in 1911. He then took positions in University of Nebraska and University of Kansas, moving to Princeton University in 1924, where he was soon given a permanent position. He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wu-Chung Hsiang
Wu-Chung Hsiang (; born 12 June 1935 in Zhejiang) is a Chinese-American mathematician, specializing in topology. Hsiang served as chairman of the Department of Mathematics at Princeton University from 1982 to 1985 and was one of the most influential topologists of the second half of the 20th century. Biography Hsiang hails from Wenzhou, Zhejiang. He received in 1957 his bachelor's degree from the National Taiwan University and in 1963 his Ph.D. under Norman Steenrod from Princeton University with thesis ''Obstructions to sectioning fibre bundles''. At Yale University he became in 1962 a lecturer, in 1963 an assistant professor, and in 1968 a full professor. At Princeton University he was a full professor from 1972 until retiring in 2006 as professor emeritus and was the department chair from 1982 to 1985. He was a visiting scholar at the Institute for Advanced Study for the academic years 1965–1966, 1971–1972, and 1979–1980. He was a visiting professor at the University o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jerome Levine
Jerome Paul Levine (May 4, 1937 – April 8, 2006) was a mathematician who contributed to the understanding of knot theory. Education and career Born in New York City, Levine received his B.S. from Massachusetts Institute of Technology in 1958, and his Ph.D. in mathematics from Princeton University in 1962, studying under Norman Steenrod. He began his career as an instructor at M.I.T., after which he spent a year at the University of Cambridge under a National Science Foundation postdoctoral fellowship. He became a professor at the University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ... in 1964, and in 1966 he left for Brandeis University. His early work helped to develop surgery theory, surgery as a powerful tool in knot theory and in geometric topolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
