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Operad
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defines an ''algebra over O'' to be a set together with concrete operations on this set which behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations. History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968 and by J. Peter May in 1972. Martin Markl, Steve Shnider, and Jim Stasheff write in their book on operads:"Operads in Algebra, Topology and Physics": Martin Markl, Steve Shnider, Jim Stasheff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koszul Duality
In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototypical example of Koszul duality was introduced by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand,. It establishes a duality between the derived category of a symmetric algebra and that of an exterior algebra, as well as the BGG correspondence, which links the stable category of finite-dimensional graded modules over an exterior algebra to the bounded derived category of coherent sheaves on projective space. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature. Koszul duality for graded modules over Koszul algebras The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space ''V'' over a field ''k'', with dual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operad Algebra
In algebra, an operad algebra is an "algebra" over an operad. It is a generalization of an associative algebra over a commutative ring ''R'', with an operad replacing ''R''. Definitions Given an operad ''O'' (say, a symmetric sequence in a symmetric monoidal ∞-category ''C''), an algebra over an operad, or ''O''-algebra for short, is, roughly, a left module over ''O'' with multiplications parametrized by ''O''. If ''O'' is a topological operad, then one can say an algebra over an operad is an ''O''-monoid object in ''C''. If ''C'' is symmetric monoidal, this recovers the usual definition. Let ''C'' be symmetric monoidal ∞-category with monoidal structure distributive over colimits. If f: O \to O' is a map of operads and, moreover, if ''f'' is a homotopy equivalence, then the ∞-category of algebras over ''O'' in ''C'' is equivalent to the ∞-category of algebras over ''O in ''C''. See also *En-ring *Homotopy Lie algebra In mathematics, in particular abstract algebra an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Ginzburg
Victor Ginzburg (born 1957) is a Russian American mathematician who works in representation theory and in noncommutative geometry. He is known for his contributions to geometric representation theory, especially, for his works on representations of quantum groups and Hecke algebras, and on the geometric Langlands program (Satake equivalence of categories). He is currently a Professor of Mathematics at the University of Chicago. Career Ginzburg received his Ph.D. at Moscow State University in 1985, under the direction of Alexandre Kirillov and Israel Gelfand. Ginzburg wrote a textbook ''Representation theory and complex geometry'' with Neil Chriss on geometric representation theory. A paper by Alexander Beilinson, Ginzburg, and Wolfgang Soergel introduced the concept of Koszul duality (cf. Koszul algebra) and the technique of "mixed categories" to representation theory. Furthermore, Ginzburg and Mikhail Kapranov developed Koszul duality theory for operads. In noncommutative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Stasheff
James Dillon Stasheff (born January 15, 1936, New York City) is an American mathematician, a professor emeritus of mathematics at the University of North Carolina at Chapel Hill. He works in algebraic topology and algebra as well as their applications to physics. Biography Stasheff did his undergraduate studies in mathematics at the University of Michigan, graduating in 1956. Stasheff then began his graduate studies at Princeton University; his notes for a 1957 course by John Milnor on characteristic classes first appeared in mimeographed form and later in 1974 in revised form book with Stasheff as a co-author. After his second year at Princeton, he moved to Oxford University on a Marshall Scholarship. Two years later in 1961, with a pregnant wife, needing an Oxford degree to get reimbursed for his return trip to the US, and yet still feeling attached to Princeton, he split his thesis into two parts (one topological, the other algebraic) and earned two doctorates, a D.Phil. from Ox ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an ''A''∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of Ω''X'', i.e. the set of based-homotopy equivalence classes of based loops in ''X'', is a group, the fundamental group ''π''1(''X''). The iterated loop spaces of ''X'' are formed by applying Ω a number of times. There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space ''X'' is the space of maps from the circle ''S''1 to ''X'' with the compact-open topology. The free loop space of ''X'' is often denoted by \mathcalX. As a functor, the free loop space construction is rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures dra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Willwacher
Thomas Hans Willwacher (born 12 April 1983) is a German mathematician and mathematical physicist working as a Professor at the Institute of Mathematics, ETH Zurich. Biography Willwacher completed his PhD at ETH Zurich in 2009 with a thesis on "Cyclic Formality", under the supervision of Giovanni Felder, Alberto Cattaneo, and Anton Alekseev. He was a Junior member of the Harvard Society of Fellows from 2010 to 2013. He joined the University of Zurich as an assistant professor in 2013, and was promoted to associate professor in 2015. Research and recognition In July 2016 Willwacher was awarded a prize from the European Mathematical Society for "his striking and important research in a variety of mathematical fields: homotopical algebra, geometry, topology and mathematical physics, including deep results related to Kontsevich's formality theorem and the relation between Kontsevich's graph complex and the Grothendieck-Teichmüller Lie algebra". Notable results of Willwacher incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology (mathematics)
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of Abelian group, abelian groups called ''homology groups.'' This operation, in turn, allows one to associate various named ''homologies'' or ''homology theories'' to various other types of mathematical objects. Lastly, since there are many homology theories for Topological space, topological spaces that produce the same answer, one also often speaks of the ''homology of a topological space''. (This latter notion of homology admits more intuitive descriptions for 1- or 2-dimensional topological spaces, and is sometimes referenced in popular mathematics.) There is also a related notion of the cohomology of a Cochain complexes, cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space. Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Homotopy Theory
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homotopy theory makes certain calculations much easier. Rational homotopy types of simply connected spaces can be identified with (isomorphism classes of) certain algebraic objects called Sullivan minimal models, which are commutative differential graded algebras over the rational numbers satisfying certain conditions. A geometric application was the theorem of Sullivan and Micheline Vigué-Poirrier (1976): every simply connected closed Riemannian manifold ''X'' whose rational cohomology ring is not generated by one element has infinitely many geometrically distinct closed geodesics. The proof used rational homotopy theory to show that the Betti numbers of the free loop space of ''X'' are unbounded. The theorem then follows from a 1969 result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
