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Chiral Homology
In mathematics, chiral homology, introduced by Alexander Beilinson and Vladimir Drinfeld, is, in their words, "a “quantum” version of (the algebra of functions on) the space of global horizontal sections of an affine \mathcal_X-scheme (i.e., the space of global solutions of a system of non-linear differential equations)." Jacob Lurie's topological chiral homology gives an analog for manifolds. See also * Ran space *Chiral Lie algebra In algebra, a chiral Lie algebra is a D-module on a curve with a certain structure of Lie algebra. It is related to an \mathcal_2-algebra via the Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to ... * Factorization homology References *{{cite book, last1=Beilinson, first1=Alexander, authorlink1=Alexander Beilinson, last2=Drinfeld, first2=Vladimir, authorlink2=Vladimir Drinfeld , title=Chiral algebras, date=2004, publisher=American Mathematical Society, isbn=0-8218-3528-9, chapter=Chap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Beilinson
Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1999 Beilinson was awarded the Ostrowski Prize with Helmut Hofer. In 2017 he was elected to the National Academy of Sciences. Work In 1978, Beilinson published a paper on coherent sheaves and several problems in linear algebra. His two-page note in the journal ''Functional Analysis and Its Applications'' was one of the papers on the study of derived categories of coherent sheaf (mathematics), sheaves. In 1981 Beilinson announced a proof of the Kazhdan–Lusztig conjectures and Jantzen conjectures with Joseph Bernstein. Independent of Beilinson and Bernstein, Jean-Luc Brylinski, Brylinski and Masaki Kashiwara, Kashiwara obtained a proof of the Kazhdan–Lusztig conjectures. However, the proof of Beilinson–Bernstein introduced a method ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Drinfeld
Vladimir Gershonovich Drinfeld ( uk, Володи́мир Ге́ршонович Дрінфельд; russian: Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a renowned mathematician from the former USSR, who emigrated to the United States and is currently working at the University of Chicago. Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence. Drinfeld introduced the notion of a quantum group (independently discovered by Michio Jimbo at the same time) and made important contributions to mathematical physics, including the ADHM construction of instantons, algebraic formalism of the quantum inverse scattering method, and the Drinfeld–Sokolov reduction in the theory of solitons. He was awarded the Fields Medal in 1990. In 2016, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Lurie
Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow. Life When he was a student in the Science, Mathematics, and Computer Science Magnet Program at Montgomery Blair High School, Lurie took part in the International Mathematical Olympiad, where he won a gold medal with a perfect score in 1994. In 1996 he took first place in the Westinghouse Science Talent Search and was featured in a front-page story in the ''Washington Times''. Lurie earned his bachelor's degree in mathematics from Harvard College in 2000 and was awarded in the same year the Morgan Prize for his undergraduate thesis on Lie algebras. He earned his Ph.D. from the Massachusetts Institute of Technology under supervision of Michael J. Hopkins, in 2004 with a thesis on derived algebraic geometry. In 2007, he became associate professor at MIT, and in 2009 he became professor at Harvard University. In 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ran Space
In mathematics, the Ran space (or Ran's space) of a topological space ''X'' is a topological space \operatorname(X) whose underlying set is the set of all empty set, nonempty finite subsets of ''X'': for a metric space ''X'' the topological space, topology is induced by the Hausdorff distance. The notion is named after Ziv Ran. Definition In general, the topology of the Ran space is generated by sets : \ for any disjoint sets, disjoint open subsets U_i \subset X, i = 1, ..., m. There is an analog of a Ran space for a Scheme (mathematics), scheme: the Ran prestack of a quasi-projective scheme ''X'' over a field (mathematics), field ''k'', denoted by \operatorname(X), is the category (mathematics), category whose object (category theory), objects are triples (R, S, \mu) consisting of a finitely generated algebra over a field, ''k''-algebra ''R'', a nonempty set ''S'' and a map of sets \mu: S \to X(R), and whose morphisms (R, S, \mu) \to (R', S', \mu') consist of a algebra homomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chiral Lie Algebra
In algebra, a chiral Lie algebra is a D-module on a curve with a certain structure of Lie algebra. It is related to an \mathcal_2-algebra via the Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generaliz .... See also * Chiral algebra * Chiral homology References * Lie algebras {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization Homology
In algebraic topology and category theory, factorization homology is a variant of topological chiral homology, motivated by an application to topological quantum field theory and cobordism hypothesis In mathematics, the cobordism hypothesis, due to John C. Baez and James Dolan, concerns the classification of extended topological quantum field theories (TQFTs). In 2008, Jacob Lurie outlined a proof of the cobordism hypothesis, though the detail ... in particular. It was introduced by David Ayala, John Francis, and Nick Rozenblyum. References * External links * Homological algebra {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
