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Segal Conjecture
Segal's Burnside ring conjecture, or, more briefly, the Segal conjecture, is a theorem in homotopy theory, a branch of mathematics. The theorem relates the Burnside ring of a finite group ''G'' to the stable cohomotopy of the classifying space ''BG''. The conjecture was made in the mid 1970s by Graeme Segal and proved in 1984 by Gunnar Carlsson. , this statement is still commonly referred to as the Segal conjecture, even though it now has the status of a theorem. Statement of the theorem The Segal conjecture has several different formulations, not all of which are equivalent. Here is a weak form: there exists, for every finite group ''G'', an isomorphism :\varprojlim \pi_S^0 \left( BG^_+ \right) \to \widehat(G). Here, lim denotes the inverse limit, S* denotes the stable cohomotopy ring, ''B'' denotes the classifying space, the superscript ''k'' denotes the ''k''-skeleton, and the subscript + denotes the addition of a disjoint basepoint. On the right-hand side, the hat denotes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has ... [...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] |
Atiyah–Segal Completion Theorem
The Atiyah–Segal completion theorem is a theorem in mathematics about equivariant K-theory in homotopy theory. Let ''G'' be a compact Lie group and let ''X'' be a ''G''-CW-complex. The theorem then states that the projection map :\pi\colon X\times EG\to X induces an isomorphism of prorings :\pi^*\colon K_G^*(X)_ \to K^*((X\times EG)/G). Here, the induced map has as domain the completion of the ''G''-equivariant K-theory of ''X'' with respect to ''I'', where ''I'' denotes the augmentation ideal of the representation ring of ''G''. In the special case of ''X'' a point, the theorem specializes to give an isomorphism K^*(BG)\cong R(G)_ between the K-theory of the classifying space of ''G'' and the completion of the representation ring. The theorem can be interpreted as giving a comparison between the geometrical process of taking the homotopy quotient of a ''G''-space, by making the action free before passing to the quotient, and the algebraic process of completing with res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004. Life Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at the University of Oxford and the University of Cambridge and in the United States at the Institute for Advanced Study. He was the President of the Royal Society (1990–1995), founding director of the Isaac Newton Institute (1990–1996), master of Trinity College, Cambridge (1990–1997), chancellor of the University of Leicester (1995–2005), and the President of the Royal Society of Edinburgh (2005–2008). From 1997 until his death, he was an honorary professor in the University of Edinburgh. Atiyah's mathematical collaborators included Raoul Bott, Friedrich Hirzebruch and Isadore Sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyahâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representable Functor
In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. From another point of view, representable functors for a category ''C'' are the functors ''given'' with ''C''. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory. Definition Let C be a locally small category and let Set be the category of sets. For each object ''A'' of C let Hom(''A'',–) be the hom functor that maps object ''X'' to the set Hom(''A'',''X''). A functor ''F'' : C → Set is said to be representable if it is naturally isomorphic to Hom(''A'',–) for some object ''A'' of C. A representation of ''F'' is a pair (''A'', Φ) where :Φ : Hom(''A'',&ndash ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with # An action of G on P, analogous to (x, g)h = (x, gh) for a product space. # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X \times G \to G that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space. A common example of a principal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis. Formal definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motivation And Interpretation
Motivation is the reason for which humans and other animals initiate, continue, or terminate a behavior at a given time. Motivational states are commonly understood as forces acting within the agent that create a disposition to engage in goal-directed behavior. It is often held that different mental states compete with each other and that only the strongest state determines behavior. This means that we can be motivated to do something without actually doing it. The paradigmatic mental state providing motivation is desire. But various other states, such as beliefs about what one ought to do or intentions, may also provide motivation. Motivation is derived from the word 'motive', which denotes a person's needs, desires, wants, or urges. It is the process of motivating individuals to take action in order to achieve a goal. The psychological elements fueling people's behavior in the context of job goals might include a desire for money. Various competing theories have been proposed con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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