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Blakers–Massey Theorem
In mathematics, the first Blakers–Massey theorem, named after Albert Blakers and William S. Massey, gave vanishing conditions for certain triad homotopy groups of spaces. Description of the result This connectivity result may be expressed more precisely, as follows. Suppose ''X'' is a topological space which is the pushout of the diagram : A\xleftarrow C \xrightarrow B, where ''f'' is an ''m''-connected map and ''g'' is ''n''-connected. Then the map of pairs : (A,C)\rightarrow (X,B) induces an isomorphism in relative homotopy groups in degrees k\le (m+n-1) and a surjection in the next degree. However the third paper of Blakers and Massey in this area determines the critical, i.e., first non-zero, triad homotopy group as a tensor product, under a number of assumptions, including some simple connectivity. This condition and some dimension conditions was relaxed in work of Ronald Brown and Jean-Louis Loday. The algebraic result implies the connectivity result, since a ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albert Blakers
Albert may refer to: Companies * Albert (supermarket), a supermarket chain in the Czech Republic * Albert Heijn, a supermarket chain in the Netherlands * Albert Market, a street market in The Gambia * Albert Productions, a record label * Albert Computers, Inc., a computer manufacturer in the 1980s Entertainment * Albert (1985 film), ''Albert'' (1985 film), a Czechoslovak film directed by František Vláčil * ''Albert'' (2015 film), a film by Karsten Kiilerich * Albert (2016 film), ''Albert'' (2016 film), an American TV movie * Albert (album), ''Albert'' (Ed Hall album), 1988 * Albert (short story), "Albert" (short story), by Leo Tolstoy * Albert (comics), a character in Marvel Comics * Albert (Discworld), Albert (''Discworld''), a character in Terry Pratchett's ''Discworld'' series * Albert (suspiria), Albert, a character in Dario Argento's 1977 film ''Suspiria'' Military * Battle of Albert (1914), a WWI battle at Albert, Somme, France * Battle of Albert (1916), a WWI battle at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected Space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Institute For Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholars, including J. Robert Oppenheimer, Albert Einstein, Hermann Weyl, John von Neumann, and Kurt Gödel, many of whom had emigrated from Europe to the United States. It was founded in 1930 by American educator Abraham Flexner, together with philanthropists Louis Bamberger and Caroline Bamberger Fuld. Despite collaborative ties and neighboring geographic location, the institute, being independent, has "no formal links" with Princeton University. The institute does not charge tuition or fees. Flexner's guiding principle in founding the institute was the pursuit of knowledge for its own sake.Jogalekar. The faculty have no classes to teach. There are no degree programs or experimental facilities at the institute. Research is never contract ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter LeFanu Lumsdaine
Peter may refer to: People * List of people named Peter, a list of people and fictional characters with the given name * Peter (given name) ** Saint Peter (died 60s), apostle of Jesus, leader of the early Christian Church * Peter (surname), a surname (including a list of people with the name) Culture * Peter (actor) (born 1952), stage name Shinnosuke Ikehata, Japanese dancer and actor * Peter (album), ''Peter'' (album), a 1993 EP by Canadian band Eric's Trip * Peter (1934 film), ''Peter'' (1934 film), a 1934 film directed by Henry Koster *Peter (2021 film), ''Peter'' (2021 film), Marathi language film * Peter (Fringe episode), "Peter" (''Fringe'' episode), an episode of the television series ''Fringe'' * Peter (novel), ''Peter'' (novel), a 1908 book by Francis Hopkinson Smith * Peter (short story), "Peter" (short story), an 1892 short story by Willa Cather Animals * Peter, the Lord's cat, cat at Lord's Cricket Ground in London * Peter (chief mouser), Chief Mouser between 1929 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Assistant
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. System comparison * ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. * Coq – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification. * HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems the logical core is a library of their programming language. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Agda (programming Language)
Agda is a dependently typed functional programming language originally developed by Ulf Norell at Chalmers University of Technology with implementation described in his PhD thesis. The original Agda system was developed at Chalmers by Catarina Coquand in 1999. The current version, originally known as Agda 2, is a full rewrite, which should be considered a new language that shares a name and tradition. Agda is also a proof assistant based on the propositions-as-types paradigm, but unlike Coq, has no separate tactics language, and proofs are written in a functional programming style. The language has ordinary programming constructs such as data types, pattern matching, records, let expressions and modules, and a Haskell-like syntax. The system has Emacs and Atom interfaces but can also be run in batch mode from the command line. Agda is based on Zhaohui Luo's unified theory of dependent types (UTT), a type theory similar to Martin-Löf type theory. Agda is named after the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (set, function, geometrical figure, number, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Type Theory
In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies. This includes, among other lines of work, the construction of homotopical and Higher category theory, higher-categorical Model (mathematical logic), models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and higher category theory; the development of mathematics within a type-theoretic foundation of mathematics, foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the Formal proof, formalization of each of these in computer proof assistants. There is a large overlap between the work referred to as homotopy type theory, and as the univalent foundations project. Although neither is precisely delinea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Rezk
Charles Waldo Rezk (born 26 January 1969) is an American mathematician, specializing in algebraic topology, category theory, and spectral algebraic geometry. Education and career Rezk matriculated at the University of Pennsylvania in 1987 and graduated there in 1991 with B.A. and M.A. in mathematics. In 1996 he received his PhD from Massachusetts Institute of Technology, MIT with thesis ''Spaces of Algebra Structures and Cohomology of Operads'' and advisor Michael J. Hopkins. At Northwestern University Rezk was a faculty member from 1996 to 2001. At the University of Illinois he was an assistant professor from 2001 to 2006 and an associate professor from 2006 to 2014 and is a full professor since 2014. He was at the Institute for Advanced Study in the fall of 1999, the spring of 2000, and the spring of 2001. He held visiting positions at MIT in 2006 and at Berkeley's Mathematical Sciences Research Institute, MSRI in 2014. Since 2015 he has been a member of the editorial board of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Pullback
In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfconsidered as an object in the homotopy category of diagrams F \in \text(\textbf^I), (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone\begin \underset(F)&: * \to \textbf\\ \underset(F)&: * \to \textbf \endwhich are objects in the homotopy category \text(\textbf^*), where * is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category \text(\textbf) since the latter homotopy functor category has functors which picks out an object in \text and a natural transformation corresponds to a continuous function of topological spaces. Note this construction can be generalized to model categories, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
