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Contact Transformation
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for ' complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Contact Structure
Standard may refer to: Symbols * Colours, standards and guidons, kinds of military signs * Standard (emblem), a type of a large symbol or emblem used for identification Norms, conventions or requirements * Standard (metrology), an object that bears a defined relationship to a unit of measure used for calibration of measuring devices * Standard (timber unit), an obsolete measure of timber used in trade * Breed standard (also called bench standard), in animal fancy and animal husbandry * BioCompute Standard, a standard for next generation sequencing * ''De facto'' standard, product or system with market dominance * Gold standard, a monetary system based on gold; also used metaphorically for the best of several options, against which the others are measured * Internet Standard, a specification ratified as an open standard by the Internet Engineering Task Force * Learning standards, standards applied to education content * Standard displacement, a naval term describing the weig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Low-dimensional Topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory. History A number of advances starting in the 1960s had the effect of emphasising low dimensions in topology. The solution by Stephen Smale, in 1961, of the Poincaré conjecture in five or more dimensions made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory. Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that sugge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential 1-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable System
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution (differential Geometry)
In differential geometry, a discipline within mathematics, a distribution on a manifold M is an assignment x \mapsto \Delta_x \subseteq T_x M of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle TM. Distributions satisfying a further integrability condition give rise to foliations, i.e. partitions of the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, e.g. integrable systems, Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology, etc. Even though they share the same name, distributions presented in this article have nothing to do with distributions in the sense of analysis. Definition Let M be a smooth manifold; a (smooth) distribution \Delta assigns to any point x \in M a vector subspace \Delta_x \subset T_xM in a smooth way. More precisely, \Delta consists in a collection \_ of vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotangent Bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories. Formal Definition Let ''M'' be a smooth manifold and let ''M''×''M'' be the Cartesian product of ''M'' with itself. The diagonal mapping Δ sends a point ''p'' in ''M'' to the point (''p'',''p'') of ''M''×''M''. The image of Δ is called the diagonal. Let \mathcal be the sheaf of germs of smooth functions on ''M''×''M'' which vanish on the diagona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, linear subspaces, flats, and affine subspaces are also called ''linear manifolds'' for emphasizing that there are also manifolds. is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a ''subspace'' when the context serves to distinguish it from other types of subspaces. Definition If ''V'' is a vector space over a field ''K'' and if ''W'' is a subset of ''V'', then ''W'' is a linear subspace of ''V'' if under the operations of ''V'', ''W'' is a vector space over ''K''. Equivalently, a nonempty subset ''W'' is a subspace of ''V'' if, whenever are elements of ''W'' and are elements of ''K'', it follows that is in ''W''. As a corollary, all vector spaces are equipped with at least two ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stein Manifold
In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry. Definition Suppose X is a complex manifold of complex dimension n and let \mathcal O(X) denote the ring of holomorphic functions on X. We call X a Stein manifold if the following conditions hold: * X is holomorphically convex, i.e. for every compact subset K \subset X, the so-called ''holomorphically convex hull'', ::\bar K = \left \, :is also a ''compact'' subset of X. * X is holomorphically separable, i.e. if x \neq y are two points in X, then there exists f \in \mathcal O(X) such that f(x) \neq f(y). Non-compact Riemann surfaces are Stein manifolds Let ''X'' be a connected, non-compact Riema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yakov Eliashberg
Yakov Matveevich Eliashberg (also Yasha Eliashberg; russian: link=no, Яков Матвеевич Элиашберг; born 11 December 1946) is an American mathematician who was born in Leningrad, USSR. Education and career Eliashberg received his PhD, entitled ''Surgery of Singularities of Smooth Mappings'', from Leningrad University in 1972, under the direction of Vladimir Rokhlin. Due to the growing anti-Semitism in the Soviet Union, from 1972 to 1979 he had to work at the Syktyvkar State University in the isolated Komi Republic. In 1980 Eliashberg returned to Leningrad and applied for a visa, but his request was denied and he became a refusenik until 1987. He was cut off from mathematical life and was prevented to work in academia, but due to a friend's intercession, he managed to secure a job in industry as the head of a computer software group. In 1988 Eliashberg managed to move to the United States, and since 1989 he has been Herald L. and Caroline L. Ritch professo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lenhard Ng
Lenhard Ng (born 1976, United States) is an American mathematician, working primarily on symplectic geometry. Ng is a professor of mathematics at Duke University. Background and education Lenhard Ng is an American of Chinese descent. His father, Jack Ng, is a professor of physics at University of North Carolina Chapel Hill. Lenhard earned his B.A. ''summa cum laude'' in Mathematics and Physics from Harvard University in three years and his Ph.D. in Mathematics from the Massachusetts Institute of Technology in 2001. He is married to Astrid Giugni. Child prodigy Ng was a child prodigy who was once thought to be the "smartest kid in America". At age 10, he earned a perfect score of 800 on the math portion of what is now called the SAT-I, a feat considered to be a “remarkable achievement” when a high school junior or senior did it. He was the youngest person to have achieved this feat on his first try. At the age of 11, he earned a perfect score on the College Board Test o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Hutchings (mathematician)
Michael Lounsbery Hutchings is an American mathematician, a professor of mathematics at the University of California, Berkeley. He is known for proving the double bubble conjecture on the shape of two-chambered soap bubbles, and for his work on circle-valued Morse theory and on embedded contact homology, which he defined. Career As an undergraduate student at Harvard University, Hutchings did an REU project with Frank Morgan at Williams College that began his interest in the mathematics of soap bubbles. He finished his undergraduate studies in 1993, and stayed at Harvard for graduate school, earning his Ph.D. in 1998 under the supervision of Clifford Taubes. After postdoctoral and visiting positions at Stanford University, the Max Planck Institute for Mathematics in Bonn, Germany, and the Institute for Advanced Study in Princeton, New Jersey, he joined the UC Berkeley faculty in 2001. His work on circle-valued Morse theory (partly in collaboration with Yi-Jen Lee) studies to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
