|
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 the sixth-largest city in the U.S. state of Ohio and the county seat of Montgomery County. A small part of the city extends into Greene County. The 2020 U.S. census estimate put the city population at 137,644, while Greater Dayton was estimated to be at 814,049 residents. The Combined Statistical Area (CSA) was 1,086,512. This makes Dayton the fourth-largest metropolitan area in Ohio and 73rd in the United States. Dayton is within Ohio's Miami Valley region, north of the Greater Cincinnati area. Ohio's borders are within of roughly 60 percent of the country's population and manufacturing infrastructure, making the Dayton area a logistical centroid for manufacturers, suppliers, and shippers. Dayton also hosts significant research and development in fields like industrial, aeronautical, and astronautical engineering that have led to many technological innovations. Much of this innovation is due in part to Wright-Patterson Air Force Base and its place in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the best universities in the world and it is among the most selective in the United States. The university is composed of an undergraduate college and five graduate research divisions, which contain all of the university's graduate programs and interdisciplinary committees. Chicago has eight professional schools: the Law School, the Booth School of Business, the Pritzker School of Medicine, the Crown Family School of Social Work, Policy, and Practice, the Harris School of Public Policy, the Divinity School, the Graham School of Continuing Liberal and Professional Studies, and the Pritzker School of Molecular Engineering. The university has additional campuses and centers in London, Paris, Beijing, Delhi, and Hong Kong, as well as in downtown ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steenrod Algebra
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. Thes ... [...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 do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cup Product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space ''X'' into a graded ring, ''H''∗(''X''), called the cohomology ring. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944. Definition In singular cohomology, the cup product is a construction giving a product on the graded cohomology ring ''H''∗(''X'') of a topological space ''X''. The construction starts with a product of cochains: if \alpha^p is a ''p''-cochain and \beta^q is a ''q''-cochain, then :(\alpha^p \smile \beta^q)(\sigma) = \alpha^p(\sigma \circ \iota_) \cdot \beta^q(\sigma \circ \iota_) where σ is a singular (''p'' + ''q'') -simplex and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The New York Times
''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid digital subscribers. It also is a producer of popular podcasts such as '' The Daily''. Founded in 1851 by Henry Jarvis Raymond and George Jones, it was initially published by Raymond, Jones & Company. The ''Times'' has won 132 Pulitzer Prizes, the most of any newspaper, and has long been regarded as a national " newspaper of record". For print it is ranked 18th in the world by circulation and 3rd in the U.S. The paper is owned by the New York Times Company, which is publicly traded. It has been governed by the Sulzberger family since 1896, through a dual-class share structure after its shares became publicly traded. A. G. Sulzberger, the paper's publisher and the company's chairman, is the fifth generation of the family to head the pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton, NJ
Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of which are now defunct. Centrally located within the Raritan Valley region, Princeton is a regional commercial hub for the Central New Jersey region and a commuter town in the New York metropolitan area.New York-Newark, NY-NJ-CT-PA Combined Statistical Area . Accessed December 5, 2020. As of the |
National Academy Of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the National Academy of Medicine (NAM). As a national academy, new members of the organization are elected annually by current members, based on their distinguished and continuing achievements in original research. Election to the National Academy is one of the highest honors in the scientific field. Members of the National Academy of Sciences serve '' pro bono'' as "advisers to the nation" on science, engineering, and medicine. The group holds a congressional charter under Title 36 of the United States Code. Founded in 1863 as a result of an Act of Congress that was approved by Abraham Lincoln, the NAS is charged with "providing independent, objective advice to the nation on matters related to science and technology. ... to provide scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miami University
Miami University (informally Miami of Ohio or simply Miami) is a public research university in Oxford, Ohio. The university was founded in 1809, making it the second-oldest university in Ohio (behind Ohio University, founded in 1804) and the 10th oldest public university (32nd overall) in the United States. The school's system comprises the main campus in Oxford, as well as regional campuses in nearby Hamilton, Middletown, and West Chester. Miami also maintains an international boarding campus, the Dolibois European Center in Differdange, Luxembourg. It is classified among "R2: Doctoral Universities – High research activity". Miami University provides a liberal arts education; it offers more than 120 undergraduate degree programs and over 60 graduate degree programs within its 8 schools and colleges in architecture, business, engineering, humanities and the sciences. In its 2021 edition, '' U.S. News & World Report'' ranked the university 103rd among universities in the ... [...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 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 gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.png "Click for more on -> The New York Times")
.JPG "Click for more on -> National Academy Of Sciences")
.jpg "Click for more on -> Miami University")
