|
Symplectic Category
In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds ''L'' into M \times N^, where the superscript minus means minus the given symplectic form (for example, the graph of a symplectomorphism; hence, minus). The notion was introduced by Alan Weinstein, according to whom "Quantization problemsHe means geometric quantization. suggest that the category of symplectic manifolds and symplectomorphisms be augmented by the inclusion of canonical relations as morphisms." The composition of canonical relations is given by a fiber product. Strictly speaking, the symplectic category is not a well-defined category (since the composition may not be well-defined) without some transversality conditions. References ;Notes ;Sources * Further reading *Victor Guillemin and Shlomo Sternberg, ''Some problems in integral geometry and some related problems in micr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation. Formal definition A diffeomorphism between two symplectic manifolds f: (M,\omega) \rightarrow (N,\omega') is called a symplectomorphism if :f^*\omega'=\omega, where f^* is the pullback of f. The symplectic diffeomorphisms from M to M are a (pseudo-)group, called the symplectomorphism group (see below). The infinitesimal version of symplectomorphisms gives the symplectic vector fields. A vector field X \in \Gamma^(TM) is called symplectic if :\mathcal_X\omega=0. Also, X is symplectic iff the flow \phi_t: M\rightarrow M of X is a symplectomorphism for every t. These vector fields build a Lie subalgebra of \Gamma^(TM). Here, \Gamma^(TM) is the set of smooth vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Weinstein
Alan David Weinstein (17 June, 1943, New York City) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson geometry. Education and career Weinstein obtained a bachelor's degree at the Massachusetts Institute of Technology in 1964. He received a PhD at University of California, Berkeley in 1967 under the direction of Shiing-Shen Chern. His dissertation was entitled "''The cut locus and conjugate locus of a Riemannian manifold''". He worked then at MIT on 1967 (as Moore instructor) and at Bonn University in 1968/69. In 1969 he became assistant professor at Berkeley, and from 1976 he is full professor. During 1978/79 he was visiting professor at Rice University. Weinstein was awarded in 1971 a Sloan Research Fellowship and in 1985 a Guggenheim Fellowship. In 1978 he was invited speaker at the International Congress of Mathematicians in Helsinki. In 1992 he was elected Fellow of the A ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Product
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often written : and comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square. The dual concept of the pullback is the '' pushout''. Universal property Explicitly, a pullback of the morphisms and consists of an object and two morphisms and for which the diagram : commutes. Moreover, the pullback must be universa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Guillemin
Victor William Guillemin (born 1937 in Boston) is an American mathematician working in the field of symplectic geometry, who has also made contributions to the fields of microlocal analysis, spectral theory, and mathematical physics. He is a tenured Professor in the Department of Mathematics at the Massachusetts Institute of Technology. His uncle Ernst Guillemin was a Professor of Electrical Engineering and Computer Science (EECS) at MIT, and his daughter Karen Guillemin is a Professor of Biology at the University of Oregon. Professional career Guillemin received a Ph.D. in mathematics from Harvard University in 1962, after earlier completing his B. A. at Harvard in 1959, as well as an M. A. at the University of Chicago in 1960. His thesis, entitled ''Theory of Finite G-Structures,'' was written under the direction of Shlomo Sternberg. He is the author or co-author of numerous books and monographs, including a widely used textbook on differential topology, written joint ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shlomo Sternberg
Shlomo Zvi Sternberg (born 1936), is an American mathematician known for his work in geometry, particularly symplectic geometry and Lie theory. Education and career Sternberg earned his PhD in 1955 from Johns Hopkins University, with a thesis entitled "''Some Problems in Discrete Nonlinear Transformations in One and Two Dimensions''", supervised by Aurel Wintner. After postdoctoral work at New York University (1956–1957) and an instructorship at University of Chicago (1957–1959), Sternberg joined the Mathematics Department at Harvard University in 1959, where he was George Putnam Professor of Pure and Applied Mathematics until 2017. Since 2017, he is Emeritus Professor at the Harvard Mathematics Department. Among other honors, Sternberg was awarded a Guggenheim fellowship in 1974 and a honorary doctorate by the University of Mannheim in 1991. He delivered the AMS in 1990 and the Hebrew University's Albert Einstein Memorial Lecture in 2006. Sternberg was elected member of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Integral Operator
In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operators as well as classical integral operators as special cases. A Fourier integral operator T is given by: :(Tf)(x)=\int_ e^a(x,\xi)\hat(\xi) \, d\xi where \hat f denotes the Fourier transform of f, a(x,\xi) is a standard symbol which is compactly supported in x and \Phi is real valued and homogeneous of degree 1 in \xi. It is also necessary to require that \det \left(\frac\right)\neq 0 on the support of ''a.'' Under these conditions, if ''a'' is of order zero, it is possible to show that T defines a bounded operator from L^ to L^. Examples One motivation for the study of Fourier integral operators is the solution operator for the initial value problem for the wave operator. Indeed, consider the following problem: : \frac\frac(t,x) = \Delta u(t,x) \quad \mathrm \quad (t,x) \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> American Journal Of Mathematics")
