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Khovanov Homology
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov. Overview To any link diagram D representing a link L, we assign the Khovanov bracket \left D \right/math>, a cochain complex of graded vector spaces. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial. Next, we normalise \left D \right/math> by a series of degree shifts (in the graded vector spaces) and height shifts (in the cochain complex) to obtain a new cochain complex C(D). The cohomology of this cochain complex turns out to be an invariant of L, and its graded Euler characteristic is the Jones polynomial of L. Definition This definition follows the formalism given in Dror Bar-Natan's 2002 paper. Let denote the ''degree shift'' operation on graded vector spaces—that is, the homogeneous comp ... [...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] |
Skein Relation
Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invariants of the knot. If two diagrams have different polynomials, they represent different knots. However, the converse is not true. Skein relations are often used to give a simple definition of knot polynomials. A skein relation gives a linear relation between the values of a knot polynomial on a collection of three links which differ from each other only in a small region. For some knot polynomials, such as the Conway, Alexander, and Jones polynomials, the relevant skein relations are sufficient to calculate the polynomial recursively. Definition A skein relationship requires three link diagrams that are identical except at one crossing. The three diagrams must exhibit the three possibilities that could occur for the two line segments at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catharina Stroppel
Catharina Stroppel (born 1971) is a German mathematician whose research concerns representation theory, low-dimensional topology, and category theory. She is a professor of mathematics at the University of Bonn, and vice-coordinator of the Hausdorff Center for Mathematics in Bonn... Education and career Stroppel earned a diploma in mathematics and theology from the University of Freiburg in 1998. She completed her doctorate, also from the University of Freiburg, in 2001, under the supervision of Wolfgang Soergel. After short-term positions at the University of Leicester and Aarhus University, she joined the University of Glasgow as a research associate in 2004, and was promoted to lecturer in 2005 and reader in 2007. In 2008 she moved to Bonn as a professor. Awards and honors In 2007, the London Mathematical Society awarded Stroppel their Whitehead Prize "for her contributions to representation theory, in particular in the framework of categorifications and its applications to ... [...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] |
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] |
Double Cover (topology)
Double, The Double or Dubble may refer to: Mathematics and computing * Multiplication by 2 * Double precision, a floating-point representation of numbers that is typically 64 bits in length * A double number of the form x+yj, where j^2=+1 * A 2-tuple, or ordered list of two elements, commonly called an ordered pair, denoted (a,b) * Double (manifold), in topology Food and drink * A drink order of two shots of hard liquor in one glass * A "double decker", a hamburger with two patties in a single bun Games * Double, action in games whereby a competitor raises the stakes ** , in contract bridge ** Doubling cube, in backgammon ** Double, doubling a blackjack bet in a favorable situation ** Double, a bet offered by UK bookmakers which combines two selections * Double, villain in the video game '' Mega Man X4'' * A kart racing game '' Mario Kart: Double Dash'' * An arcade action game ''Double Dragon'' Sports * Double (association football), the act of a winning a division and pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zoltán Szabó (mathematician)
Zoltán Szabó (born November 24, 1965) is a professor of mathematics at Princeton University known for his work on Heegaard Floer homology. Education and career Szabó received his BA from Eötvös Loránd University in Budapest, Hungary in 1990, and he received his PhD from Rutgers University in 1994. Together with Peter Ozsváth, Szabó created Heegaard Floer homology, a homology theory for 3-manifolds. For this contribution to the field of topology, Ozsváth and Szabó were awarded the 2007 Oswald Veblen Prize in Geometry. In 2010, he was elected honorary member of the Hungarian Academy of Sciences The Hungarian Academy of Sciences ( , MTA) is Hungary’s foremost and most prestigious learned society. Its headquarters are located along the banks of the Danube in Budapest, between Széchenyi rakpart and Akadémia utca. The Academy's primar .... Selected publications *. *.Homology for Knots and Links'' American Mathematical Society, (2015) References External links ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Ozsváth
Peter Steven Ozsváth (born October 20, 1967) is a professor of mathematics at Princeton University. He created, along with Zoltán Szabó, Heegaard Floer homology, a homology theory for 3-manifolds. Education Ozsváth received his PhD from Princeton in 1994 under the supervision of John Morgan; his dissertation was entitled ''On Blowup Formulas For SU(2) Donaldson Polynomials''. Awards In 2007, Ozsváth was one of the recipients of the Oswald Veblen Prize in Geometry. In 2008 he was named a Guggenheim Fellow. In July 2017, he was a plenary lecturer in the Mathematical Congress of the Americas. He was elected a member of the National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ... in 2018. Selected publications * *Homology for Knots and Links'' America ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Discovery and motivation Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf and found himself faced with the problem of computing sheaf cohomology. To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence. This gave a relation between cohomology groups of a sheaf and cohomology groups of the pushforward of the sheaf. The relation involved an infinite process. Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomology of the cohomology. This was still not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor Conjecture (knot Theory)
In knot theory, the Milnor conjecture says that the slice genus of the (p, q) torus knot is :(p-1)(q-1)/2. It is in a similar vein to the Thom conjecture. It was first proved by gauge theoretic methods by Peter Kronheimer and Tomasz Mrowka. Jacob Rasmussen later gave a purely combinatorial proof using Khovanov homology In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov. Overv ..., by means of the s-invariant.. References Geometric topology Knot theory 4-manifolds Conjectures that have been proved {{knottheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomasz Mrowka
Tomasz Mrowka (born September 8, 1961) is an American mathematician specializing in differential geometry and gauge theory. He is the Singer Professor of Mathematics and former head of the Department of Mathematics at the Massachusetts Institute of Technology. Mrowka is the son of Polish mathematician , and is married to MIT mathematics professor Gigliola Staffilani. Career A 1983 graduate of the Massachusetts Institute of Technology, he received the PhD from the University of California, Berkeley in 1988 under the direction of Clifford Taubes and Robion Kirby. He joined the MIT mathematics faculty as professor in 1996, following faculty appointments at Stanford University and at the California Institute of Technology (professor 1994–96). At MIT, he was the Simons Professor of Mathematics from 2007–2010. Upon Isadore Singer's retirement in 2010 the name of the chair became the Singer Professor of Mathematics which Mrowka held until 2017. He was named head of the Departmen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Kronheimer
Peter Benedict Kronheimer (born 1963) is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University and former chair of the mathematics department. Education Kronheimer attended the City of London School. He completed his DPhil at Oxford University under the direction of Michael Atiyah. He has had a long association with Merton College, the oldest of the constituent colleges of Oxford University, being an undergraduate, graduate, and full fellow of the college. Career Kronheimer's early work was on gravitational instantons, in particular the classification of hyperkähler 4-manifolds with asymptotical locally Euclidean geometry (ALE spaces), leading to the papers "The construction of ALE spaces as hyper-Kähler quotients" and "A Torelli-type theorem for gravitational instantons." He and Hiraku Nakajima gave a construction of instantons on A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
