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Bertram Kostant
Bertram Kostant (May 24, 1928 – February 2, 2017) was an American mathematician who worked in representation theory, differential geometry, and mathematical physics. Early life and education Kostant grew up in New York City, where he graduated from Stuyvesant High School in 1945. He went on to obtain an undergraduate degree in mathematics from Purdue University in 1950. He earned his Ph.D. from the University of Chicago in 1954, under the direction of Irving Segal, where he wrote a dissertation on representations of Lie groups. Career in mathematics After time at the Institute for Advanced Study, Princeton University, and the University of California, Berkeley, he joined the faculty at the Massachusetts Institute of Technology, where he remained until his retirement in 1993. Kostant's work has involved representation theory, Lie groups, Lie algebras, homogeneous spaces, differential geometry and mathematical physics, particularly symplectic geometry. He has given several ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Research Institute Of Oberwolfach
The Oberwolfach Research Institute for Mathematics (german: Mathematisches Forschungsinstitut Oberwolfach) is a center for mathematical research in Oberwolfach, Germany. It was founded by mathematician Wilhelm Süss in 1944. It organizes weekly workshops on diverse topics where mathematicians and scientists from all over the world come to do collaborative research. The Institute is a member of the Leibniz Association, funded mainly by the German Federal Ministry of Education and Research and by the state of Baden-Württemberg. It also receives substantial funding from the ''Friends of Oberwolfach'' foundation, from the ''Oberwolfach Foundation'' and from numerous donors. History The Oberwolfach Research Institute for Mathematics (MFO) was founded as the ''Reich Institute of Mathematics'' (German: ''Reichsinstitut für Mathematik'') on 1 September 1944. It was one of several research institutes founded by the Nazis in order to further the German war effort, which at that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moss Sweedler
Moss Eisenberg Sweedler (born 29 April 1942, in Brooklyn) is an American mathematician, known for Sweedler's Hopf algebra, Sweedler's notation, measuring coalgebras, and his proof, with Harry Prince Allen, of a conjecture of Nathan Jacobson. Education and career Sweedler received his Ph.D. from the Massachusetts Institute of Technology in 1965. His thesis, ''Commutative Hopf Algebras with Antipode,'' was written under the direction of thesis advisor Bertram Kostant. Sweedler wrote ''Hopf Algebras'' (1969), which became the standard reference book on Hopf algebras. He, with Harry P. Allen, used Hopf algebras to prove in 1969 a famous 25-year-old conjecture of Jacobson about the forms of generalized Witt algebras over algebraically closed fields of finite characteristic. From 1965 to the mid 1980s Sweeder worked on commutative algebra and related disciplines. Since the mid 1980s Sweedler has worked primarily on computer algebra. His research resulted in his position as director of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stuyvesant High School
Stuyvesant High School (pronounced ), commonly referred to among its students as Stuy (pronounced ), is a State school, public university-preparatory school, college-preparatory, Specialized high schools in New York City, specialized high school in New York City, United States. Operated by the New York City Department of Education, these specialized schools offer Tuition payments, tuition-free accelerated academics to city residents. Stuyvesant was established as an all-boys school in the East Village, Manhattan, East Village of Manhattan in 1904. An entrance examination was mandated for all applicants starting in 1934, and the school started accepting female students in 1969. Stuyvesant moved to its current location at Battery Park City in 1992 because the student body had become too large to be suitably accommodated in the original campus. The old building now houses several high schools. Admission to Stuyvesant involves passing the Specialized High Schools Admissions Test. Eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New York City
New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the List of United States cities by population density, most densely populated major city in the United States, and is more than twice as populous as second-place Los Angeles. New York City lies at the southern tip of New York (state), New York State, and constitutes the geographical and demographic center of both the Northeast megalopolis and the New York metropolitan area, the largest metropolitan area in the world by urban area, urban landmass. With over 20.1 million people in its metropolitan statistical area and 23.5 million in its combined statistical area as of 2020, New York is one of the world's most populous Megacity, megacities, and over 58 million people live within of the city. New York City is a global city, global Culture of New ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wigner Medal
The International Colloquium on Group Theoretical Methods in Physics is an academic conference devoted to applications of group theory to physics. It was founded in 1972 by Henri Bacry and Aloysio Janner. It hosts a colloquium every two years. The ICGTMP is led by a standing committee, which helps select winners for the two major awards presented at the conference: the Wigner Medal and the Weyl Prize. Wigner Medal The Wigner Medal is an award designed "to recognize outstanding contributions to the understanding of physics through Group Theory". The Wigner Medal is administered by The Group Theory and Fundamental Physics Foundation, a publicly supported organization. Donations are tax-deductible as provided pursuant to the provisions of Section 170 of the Internal Revenue Code, a federal code of the United States. The award was first presented in 1978 to Eugene Wigner, and was first awarded at the Integrative Conference on Group Theory and Mathematical Physics. List of conferen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hochschild Homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, and extended to algebras over more general rings by . Definition of Hochschild homology of algebras Let ''k'' be a field, ''A'' an associative ''k''-algebra, and ''M'' an ''A''-bimodule. The enveloping algebra of ''A'' is the tensor product A^e=A\otimes A^o of ''A'' with its opposite algebra. Bimodules over ''A'' are essentially the same as modules over the enveloping algebra of ''A'', so in particular ''A'' and ''M'' can be considered as ''Ae''-modules. defined the Hochschild homology and cohomology group of ''A'' with coefficients in ''M'' in terms of the Tor functor and Ext functor by : HH_n(A,M) = \operatorname_n^(A, M) : HH^n(A,M) = \operatorname^n_(A, M) Hochschild complex Let ''k'' be a ring, ''A'' an associative ''k''-algebra th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Quantization
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in. Origins One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927. Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. In 1946, H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kostant Polynomial
In mathematics, the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring of polynomials over the ring of polynomials invariant under the finite reflection group of a root system. Background If the reflection group ''W'' corresponds to the Weyl group of a compact semisimple group ''K'' with maximal torus ''T'', then the Kostant polynomials describe the structure of the de Rham cohomology of the generalized flag manifold ''K''/''T'', also isomorphic to ''G''/''B'' where ''G'' is the complexification of ''K'' and ''B'' is the corresponding Borel subgroup. Armand Borel showed that its cohomology ring is isomorphic to the quotient of the ring of polynomials by the ideal generated by the invariant homogeneous polynomials of positive degree. This ring had already been considered by Claude Chevalley in establishing the foundations of the cohomology of compact Lie groups and their homogeneous spaces with André Weil, Jean-Louis Koszul and Henri Cartan; the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
