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Subfactor
In the theory of von Neumann algebras, a subfactor of a factor M is a subalgebra that is a factor and contains 1 . The theory of subfactors led to the discovery of the Jones polynomial in knot theory. Index of a subfactor Usually M is taken to be a factor of type _1 , so that it has a finite trace. In this case every Hilbert space module H has a dimension \dim_M(H) which is a non-negative real number or + \infty . The index :N of a subfactor N is defined to be \dim_N(L^2(M)) . Here L^2(M) is the representation of N obtained from the GNS construction of the trace of M . Jones index theorem This states that if N is a subfactor of M (both of type _1 ) then the index :N/math> is either of the form 4 \cos(\pi /n)^2 for n = 3,4,5,... , or is at least 4 . All these values occur. The first few values of 4 \cos(\pi /n)^2 are 1, 2, (3 + \sqrt)/2 = 2.618..., 3, 3.247..., ... Basic construction Suppose that N is a subfactor of M , and that both ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Algebra
In mathematics, planar algebras first appeared in the work of Vaughan Jones on the Subfactor#Standard invariant, standard invariant of a II-1 subfactor, II1 subfactor. They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect to tangle (mathematics), tangle composition. Any subfactor planar algebra provides a family of unitary representations of Thompson groups. Any finite group (and quantum generalization) can be encoded as a planar algebra. Definition The idea of the planar algebra is to be a diagrammatic axiomatization of the Subfactor#Standard invariant, standard invariant. Planar tangle A (shaded) planar tangle is the data of finitely many ''input'' disks, one ''output'' disk, non-intersecting strings giving an even number, say 2n , intervals per disk and one \star-marked interval per disk. image:Tangle.png, 200px Here, the mark ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sorin Popa
Sorin Teodor Popa (born 24 March 1953) is a Romanian American mathematician working on operator algebras. He is a professor at the University of California, Los Angeles. He was elected a Member of the National Academy of Sciences in 2025. Biography Popa earned his PhD from the University of Bucharest in 1983 under the supervision of Dan-Virgil Voiculescu, with thesis ''Studiul unor clase de subalgebre ale C^*-algebrelor''. He has advised 15 doctoral students at UCLA, including Adrian Ioana. Honors and awards In 1990, Popa was an invited speaker at the International Congress of Mathematicians (ICM) in Kyoto, where he gave a talk on "Subfactors and Classifications in von Neumann algebras". He was a Guggenheim Fellow in 1995. In 2006, he gave a plenary lecture at the ICM in Madrid on "Deformation and Rigidity for group actions and Von Neumann Algebras". In 2009, he was awarded the Ostrowski Prize, and in 2010 the E. H. Moore Prize. He is one of the inaugural fellows of the American ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable functions. *The algebra \mathcal B(\mathcal H) of all bounded operators on a Hilbert space \mathcal H is a von Neuma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current (2023) managing editors are Jean-Benoît Bost (University of Paris-Sud) and Wilhelm Schlag (Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Academic journals established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Group
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see ). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Emil Artin, Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see ); and in monodromy invariants of algebraic geometry. Introduction In this introduction let ; the generalization to other values of will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temperley–Lieb Algebra
In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrix, transfer matrices, invented by Harold Neville Vazeille Temperley, Neville Temperley and Elliott H. Lieb, Elliott Lieb. It is also related to integrable models, knot theory and the braid group, braid groups, quantum groups and subfactors of von Neumann algebras. Structure Generators and relations Let R be a commutative ring and fix \delta \in R. The Temperley–Lieb algebra TL_n(\delta) is the algebra (ring theory), R-algebra generated by the elements e_1, e_2, \ldots, e_, subject to the Jones relations: *e_i^2 = \delta e_i for all 1 \leq i \leq n-1 *e_i e_ e_i = e_i for all 1 \leq i \leq n-2 *e_i e_ e_i = e_i for all 2 \leq i \leq n-1 *e_i e_j = e_j e_i for all 1 \leq i,j \leq n-1 such that , i-j, \neq 1 Using these relations, any product of generators e_i can be brought to Jones' normal form: : E= \big(e_e_\cdots e_\big)\big(e_e_\cdots e_\big)\cdots\big(e_e_\cdots e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutant
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of elements g\in G such that conjugation by g leaves each element of ''S'' fixed. The normalizer of ''S'' in ''G'' is the set of elements \mathrm_G(S) of ''G'' that satisfy the weaker condition of leaving the set S \subseteq G fixed under conjugation. The centralizer and normalizer of ''S'' are subgroups of ''G''. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets ''S''. Suitably formulated, the definitions also apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the multiplication of the ring (a semigroup operation). The centralizer of a subset of a ring ''R'' is a subring of ''R''. This article also deals with centralizers and nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry. He was a professor at the , , Ohio State University and Vanderbilt University. He was awarded the Fields Medal in 1982. Career Alain Connes attended high school at in Marseille, and was then a student of the classes préparatoires in . Between 1966 and 1970 he studied at École normale supérieure in Paris, and in 1973 he obtained a PhD from Pierre and Marie Curie University, under the supervision of Jacques Dixmier. From 1970 to 1974 he was research fellow at the French National Centre for Scientific Research and during 1975 he held a visiting position at Queen's University at Kingston in Canada. In 1976 he returned to France and worked as professor at Pierre and Marie Curie University until 1980 and at CNRS between 1981 and 1984. Moreover, since 1979 he holds the Léon Motchane Chair at IHES. From 1984 until his retir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factor (functional Analysis)
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of Bounded linear operator, bounded operators on a Hilbert space that is Closed set, closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of operator theory, single operators, group representations, ergodic theory and quantum mechanics. His von Neumann double commutant theorem, double commutant theorem shows that the Mathematical analysis, analytic definition is equivalent to a purely abstract algebra, algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica
''Acta Mathematica'' is a peer-reviewed open-access scientific journal covering research in all fields of mathematics. According to Cédric Villani, this journal is "considered by many to be the most prestigious of all mathematical research journals".. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 4.273, ranking it 5th out of 330 journals in the category "Mathematics". Publication history The journal was established by Gösta Mittag-Leffler in 1882 and is published by Institut Mittag-Leffler, a research institute for mathematics belonging to the Royal Swedish Academy of Sciences. The journal was printed and distributed by Springer from 2006 to 2016. Since 2017, Acta Mathematica has been published electronically and in print by International Press. Its electronic version is open access without publishing fees. Poincaré episode The journal's "most famous episode" (according to Villani) concerns Henri Poincaré, who won a prize offered in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
