|
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] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Number
The Catalan numbers are a sequence of natural numbers that occur in various Enumeration, counting problems, often involving recursion, recursively defined objects. They are named after Eugène Charles Catalan, Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The -th Catalan number can be expressed directly in terms of the central binomial coefficients by :C_n = \frac = \frac \qquad\textn\ge 0. The first Catalan numbers for are : . Properties An alternative expression for is :C_n = - for n\ge 0\,, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a #Second proof, proof of the correctness of the formula. Another alternative expression is :C_n = \frac \,, which can be directly interpreted in terms of the cycle lemma; see below. The Catalan numbers satisfy the recurr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stefaan Vaes
Stefaan Vaes (born February 29, 1976, in Herentals, Belgium) is a Belgian mathematician. Vaes studied mathematics at KU Leuven with a diploma in 1998 and a PhD in 2001 with thesis advisor Alfons Van Daele and thesis Locally Compact Quantum Groups. As a postdoc, he worked from 1998 to 2002 at KU Leuven and from 1998 to 2002 in Paris, where he did research for CNRS. In 2002, he began part-time teaching at KU Leuven, where he became an associate professor in 2006 and a full professor in 2009. He was a visiting professor in 2009 at Pierre and Marie Curie University (Paris VI) and in 2011 at Paris Diderot University (Paris VII) (where he habilitated in 2004). In 2005, he held the ''Peccot Chair'' at the Collège de France. His research deals with Von Neumann algebras and quantum groups. In 2010, Vaes was an invited speaker with a talk on ''Rigidity for von Neumann algebras and their invariants'' at the International Congress of Mathematicians in Hyderabad. In 2012, he was elected a Fe ... [...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] |
Stephen Bigelow
Stephen John Bigelow is an Australian mathematician and professor of mathematics at the University of California, Santa Barbara.. He is known for his proof that braid groups are linear, concurrently with and independently of another proof by Daan Krammer. Bigelow earned bachelor's and master's degrees in 1992 and 1994 from the University of Melbourne. He completed his PhD in 2000 from the University of California, Berkeley under the joint supervision of Robion Kirby and Andrew Casson. He returned to Melbourne for two years as a research fellow before joining the UCSB faculty in 2002. Bigelow was an invited speaker at the International Congress of Mathematicians in 2002, speaking on representations of braid groups. He was a Sloan Research Fellow for 2002–2006. In 2012 he was designated as one of the inaugural fellows of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uffe Haagerup
Uffe Valentin Haagerup (19 December 1949 – 5 July 2015) was a mathematician from Denmark. Biography Uffe Haagerup was born in Kolding, but grew up on the island of Funen, in the small town of Fåborg. The field of mathematics had his interest from early on, encouraged and inspired by his older brother. In fourth grade Uffe was doing trigonometric and logarithmic calculations. He graduated as a student from Svendborg Gymnasium in 1968, whereupon he relocated to Copenhagen and immediately began his studies of mathematics and physics at the University of Copenhagen, again inspired by his older brother who also studied the same subjects at the same university. Early university studies in Einstein's general theory of relativity and quantum mechanics, sparked a lasting interest in the mathematical field of operator algebra, in particular Von Neumann algebra and Tomita–Takesaki theory. In 1974 he received his Candidate's degree ( cand. scient.) from the University of Copenha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proc
Proc may refer to: * Proč, a village in eastern Slovakia * '' Proč?'', a 1987 Czech film * procfs or proc filesystem, a special file system (typically mounted to ) in Unix-like operating systems for accessing process information * Protein C (PROC) * Proc, a term in video game terminology * Procedures or process, in the programming language ALGOL 68 * People's Republic of China, the formal name of China * the official acronym for the Canadian House of Commons Standing Committee on Procedure and House Affairs * People's Republic of the Congo The People's Republic of the Congo () was a Marxist–Leninist socialist state that existed in the Republic of the Congo from 1969 to 1992. The People's Republic of the Congo was founded in December 1969 as the first Marxist-Leninist state ... * Pro*C, a programming language {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vaughan F
Vaughan ( ) (2022 population 344,412) is a city in Ontario, Canada. It is located in the Regional Municipality of York, just north of Toronto. Vaughan was the fastest-growing municipality in Canada between 1996 and 2006 with its population increasing by 80.2% during this time period and having nearly doubled in population since 1991. In 2021, the population of Vaughan was 323,103. It is the fifth-largest city in the Greater Toronto Area, and the 17th-largest city in Canada. Toponymy The township was named after Benjamin Vaughan, a British commissioner who signed a peace treaty with the United States in 1783. History In the late pre-contact period, the Huron- Wendat people populated what is today Vaughan. The Skandatut ancestral Wendat village overlooked the east branch of the Humber River (Pine Valley Drive) and was once home to approximately 2,000 Huron in the sixteenth century. The site is close to a Huron ossuary (mass grave) uncovered in Kleinburg in 1970, and one kilomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alice Guionnet
Alice Guionnet (born 24 May 1969) is a French mathematician known for her work in probability theory, in particular on large random matrices. Biography Guionnet entered the École Normale Supérieure (Paris) in 1989. She earned her PhD in 1995 under the supervision of Gérard Ben Arous at University of Paris-Sud. Focuses of her academic research can be viewed in her thesis, ''Dynamique de Langevin d'un verre de spins'' (Langevin Dynamics of spin glass). She has held positions at the Courant Institute, Berkeley, MIT, and ENS (Paris). She is currently a Director of Research at ENS de Lyon. Works Alice Guionnet is known for her work on large random matrices. In this context, she established principles of large deviations for the empirical measurements of the eigenvalues of large random matrices with Gérard Ben Arous and Ofer Zeitouni, applied the theory of concentration of measure, initiated the rigorous study of matrices with a heavy tail, and obtained the convergence o ... [...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] |
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] |
