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Ricci Flow
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation. The Ricci flow, so named for the presence of the Ricci tensor in its definition, was introduced by Richard Hamilton, who used it through the 1980s to prove striking new results in Riemannian geometry. Later extensions of Hamilton's methods by various authors resulted in new applications to geometry, including the resolution of the differentiable sphere conjecture by Simon Brendle and Richard Schoen. Following Shing-Tung Yau's suggestion that the singularities of solutions of the Ricci flow could identify the topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Flow
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation. The Ricci flow, so named for the presence of the Ricci tensor in its definition, was introduced by Richard Hamilton, who used it through the 1980s to prove striking new results in Riemannian geometry. Later extensions of Hamilton's methods by various authors resulted in new applications to geometry, including the resolution of the differentiable sphere conjecture by Simon Brendle and Richard Schoen. Following Shing-Tung Yau's suggestion that the singularities of solutions of the Ricci flow could identify the topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by Henri Poincaré in 1904, the Grigori Perelman's theorem concerns spaces that locally look like ordinary Euclidean space, three-dimensional space but which are finite in extent. Poincaré hypothesized that if such a space has the additional property that each path (topology), loop in the space can be continuously tightened to a point, then it is necessarily a 3-sphere, three-dimensional sphere. Attempts to resolve the conjecture drove much progress in the field of geometric topology during the 20th century. The Perelman's proof built upon Richard S. Hamilton's ideas of using the Ricci flow to solve the problem. By developing a number of breakthrough new techniques and results in the theory of Ricci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash–Moser Theorem
In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required solution mapping for the linearized problem is not bounded. Introduction In contrast to the Banach space case, in which the invertibility of the derivative at a point is sufficient for a map to be locally invertible, the Nash–Moser theorem requires the derivative to be invertible in a neighborhood. The theorem is widely used to prove local existence for non-linear partial differential equations in spaces of smooth functions. It is particularly useful when the inverse to the derivative "loses" derivatives, and therefore the Banach space implicit function theorem cannot be used. History The Nash–Moser theorem traces back to , who proved the theorem in the special case of the isometric embedding problem. It is clear from his paper that his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor. The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renormalization Group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying force laws (codified in a quantum field theory) as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the uncertainty principle. A change in scale is called a scale transformation. The renormalization group is intimately related to ''scale invariance'' and ''conformal invariance'', symmetries in which a system appears the same at all scales (so-called self-similarity). As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system. In so-called renormalizable theories, the system at one scale will generally be seen to consist of self-similar copies of itself when viewed at a smaller sca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states (also called Quantum, quanta) of their underlying quantum field (physics), fields, which are more fundamental than the particles. The equation of motion of the particle is determined by minimization of the Lagrangian, a functional of fields associated with the particle. Interactions between particles are described by interaction terms in the Lagrangian (field theory), Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory (quantum mechanics), perturbation theory in quantum mechanics. History Quantum field theory emerged from the wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations. Definitions Let L be linear differential operator of order ''m'' on a domain \Omega in R''n'' given by Lu = \sum_ a_\alpha(x)\partial^\alpha u where \alpha = (\alpha_1, \dots, \alpha_n) denotes a multi-index, and \partial^\alpha u = \partial^_1 \cdots \partial_n^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Curvature Tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is ''flat'', i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (linear Algebra)
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sectional Curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a point ''p'' of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σ''p'' as a tangent plane at ''p'', obtained from geodesics which start at ''p'' in the directions of σ''p'' (in other words, the image of σ''p'' under the exponential map at ''p''). The sectional curvature is a real-valued function on the 2-Grassmannian fiber bundle, bundle over the manifold. The sectional curvature determines the Riemann curvature tensor, curvature tensor completely. Definition Given a Riemannian manifold and two linearly independent tangent vectors at the same point, ''u'' and ''v'', we can define :K(u,v)= Here ''R'' is the Riemann curvature tensor, defined here by the convention R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Bilinear Form
In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function B that maps every pair (u,v) of elements of the vector space V to the underlying field such that B(u,v)=B(v,u) for every u and v in V. They are also referred to more briefly as just symmetric forms when "bilinear" is understood. Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond to symmetric matrices given a basis for ''V''. Among bilinear forms, the symmetric ones are important because they are the ones for which the vector space admits a particularly simple kind of basis known as an orthogonal basis (at least when the characteristic of the field is not 2). Given a symmetric bilinear form ''B'', the function is the associated quadratic form on the vector space. Moreover, if the characteristic of ... [...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] |
