|
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] |
Soviet Union
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen national republics; in practice, both its government and its economy were highly centralized until its final years. It was a one-party state governed by the Communist Party of the Soviet Union, with the city of Moscow serving as its capital as well as that of its largest and most populous republic: the Russian SFSR. Other major cities included Leningrad (Russian SFSR), Kiev (Ukrainian SSR), Minsk ( Byelorussian SSR), Tashkent (Uzbek SSR), Alma-Ata (Kazakh SSR), and Novosibirsk (Russian SFSR). It was the largest country in the world, covering over and spanning eleven time zones. The country's roots lay in the October Revolution of 1917, when the Bolsheviks, under the leadership of Vladimir Lenin, overthrew the Russian Provisional Government ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ostrowski Prize
The Ostrowski Prize is a mathematics award given every odd year for outstanding mathematical achievement judged by an international jury from the universities of Basel, Jerusalem, Waterloo and the academies of Denmark and the Netherlands. Alexander Ostrowski, a longtime professor at the University of Basel, left his estate to the foundation in order to establish a prize for outstanding achievements in pure mathematics and the foundations of numerical mathematics. It currently carries a monetary award of 100,000 Swiss francs. Recipients * 1989: Louis de Branges (France / United States) * 1991: Jean Bourgain (Belgium) * 1993: Miklós Laczkovich (Hungary) and Marina Ratner (Russia / United States) * 1995: Andrew J. Wiles (UK) * 1997: Yuri V. Nesterenko (Russia) and Gilles I. Pisier (France) * 1999: Alexander A. Beilinson (Russia / United States) and Helmut H. Hofer (Switzerland / United States) * 2001: Henryk Iwaniec (Poland / United States) and Peter Sar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jantzen Conjectures
In representation theory, a Jantzen filtration is a filtration of a Verma module of a semisimple Lie algebra, or a Weyl module of a reductive algebraic group of positive characteristic. Jantzen filtrations were introduced by . Jantzen filtration for Verma modules If ''M''(λ) is a Verma module of a semisimple Lie algebra with highest weight λ, then the Janzen filtration is a decreasing filtration :M(\lambda)=M(\lambda)^0\supseteq M(\lambda)^1\supseteq M(\lambda)^2\supseteq\cdots. It has the following properties: *''M''(λ)1=''N''(λ), the unique maximal proper submodule of ''M''(λ) *The quotients ''M''(λ)''i''/''M''(λ)''i''+1 have non-degenerate contravariant bilinear forms. * The Jantzen sum formula holds: :\sum_\text(M(\lambda)^i) = \sum_\text(M(s_\alpha \cdot \lambda)) : where \text(\cdot) denotes the formal character. References * * *{{Citation , last1=Jantzen , first1=Jens Carsten , title=Moduln mit einem höchsten Gewicht , publisher=Springer-Verlag Springer S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Categories
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X-modules which has a local presentation, that is, every point in X has an open neighborhood U in which there is an exac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
National Academy Of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, 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 National Academy of Medicine (NAM). As a national academy, new members of the organization are elected annually by current members, based on their distinguished and continuing achievements in original research. Election to the National Academy is one of the highest honors in the scientific field. Members of the National Academy of Sciences serve '' pro bono'' as "advisers to the nation" on science, engineering, and medicine. The group holds a congressional charter under Title 36 of the United States Code. Founded in 1863 as a result of an Act of Congress that was approved by Abraham Lincoln, the NAS is charged with "providing independent, objective advice to the nation on matters related to science and technology. ... to provide scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Hofer
Helmut Hermann W. Hofer (born February 28, 1956) is a German-American mathematician, one of the founders of the area of symplectic topology. He is a member of the National Academy of Sciences, and the recipient of the 1999 Ostrowski Prize and the 2013 Heinz Hopf Prize. Since 2009, he is a faculty member at the Institute for Advanced Study in Princeton, New Jersey. He currently works on symplectic geometry, dynamical systems, and partial differential equations. His contributions to the field include Hofer geometry. Hofer was elected to the American Academy of Arts and Sciences in 2020. He was an invited speaker at the International Congress of Mathematicians (ICM) in 1990 in Kyoto and a plenary speaker at the ICM in 1998 in Berlin. Selected publications * * * * * * * * Notes External links *Oberwolfach photos of Helmut Hofer {{DEFAULTSORT:Hofer, Helmut Geometers 20th-century German mathematicians 21st-century German mathematicians Institute for Advanced Study f ... [...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] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.JPG "Click for more on -> National Academy Of Sciences")
