Showing papers on "Equivariant map published in 1992"

Heat Kernels and Dirac Operators

Abstract: 1 Background on Differential Geometry.- 1.1 Fibre Bundles and Connections.- 1.2 Riemannian Manifolds.- 1.3 Superspaces.- 1.4 Superconnections.- 1.5 Characteristic Classes.- 1.6 The Euler and Thorn Classes.- 2 Asymptotic Expansion of the Heat Kernel.- 2.1 Differential Operators.- 2.2 The Heat Kernel on Euclidean Space.- 2.3 Heat Kernels.- 2.4 Construction of the Heat Kernel.- 2.5 The Formal Solution.- 2.6 The Trace of the Heat Kernel.- 2.7 Heat Kernels Depending on a Parameter.- 3 Clifford Modules and Dirac Operators.- 3.1 The Clifford Algebra.- 3.2 Spinors.- 3.3 Dirac Operators.- 3.4 Index of Dirac Operators.- 3.5 The Lichnerowicz Formula.- 3.6 Some Examples of Clifford Modules.- 4 Index Density of Dirac Operators.- 4.1 The Local Index Theorem.- 4.2 Mehler's Formula.- 4.3 Calculation of the Index Density.- 5 The Exponential Map and the Index Density.- 5.1 Jacobian of the Exponential Map on Principal Bundles.- 5.2 The Heat Kernel of a Principal Bundle.- 5.3 Calculus with Grassmann and Clifford Variables.- 5.4 The Index of Dirac Operators.- 6 The Equivariant Index Theorem.- 6.1 The Equivariant Index of Dirac Operators.- 6.2 The Atiyah-Bott Fixed Point Formula.- 6.3 Asymptotic Expansion of the Equivariant Heat Kernel.- 6.4 The Local Equivariant Index Theorem.- 6.5 Geodesic Distance on a Principal Bundle.- 6.6 The heat kernel of an equivariant vector bundle.- 6.7 Proof of Proposition 6.13.- 7 Equivariant Differential Forms.- 7.1 Equivariant Characteristic Classes.- 7.2 The Localization Formula.- 7.3 Bott's Formulas for Characteristic Numbers.- 7.4 Exact Stationary Phase Approximation.- 7.5 The Fourier Transform of Coadjoint Orbits.- 7.6 Equivariant Cohomology and Families.- 7.7 The Bott Class.- 8 The Kirillov Formula for the Equivariant Index.- 8.1 The Kirillov Formula.- 8.2 The Weyl and Kirillov Character Formulas.- 8.3 The Heat Kernel Proof of the Kirillov Formula.- 9 The Index Bundle.- 9.1 The Index Bundle in Finite Dimensions.- 9.2 The Index Bundle of a Family of Dirac Operators.- 9.3 The Chern Character of the Index Bundle.- 9.4 The Equivariant Index and the Index Bundle.- 9.5 The Case of Varying Dimension.- 9.6 The Zeta-Function of a Laplacian.- 9.7 The Determinant Line Bundle.- 10 The Family Index Theorem.- 10.1 Riemannian Fibre Bundles.- 10.2 Clifford Modules on Fibre Bundles.- 10.3 The Bismut Superconnection.- 10.4 The Family Index Density.- 10.5 The Transgression Formula.- 10.6 The Curvature of the Determinant Line Bundle.- 10.7 The Kirillov Formula and Bismut's Index Theorem.- References.- List of Notation.

The Asymptotics of Rousseeuw's Minimum Volume Ellipsoid Estimator

Abstract: Rousseeuw's minimum volume estimator for multivariate location and dispersion parameters has the highest possible breakdown point for an affine equivariant estimator In this paper we establish that it satisfies a local Holder condition of order $1/2$ and converges weakly at the rate of $n^{-1/3}$ to a non-Gaussian distribution

Orientation and string structures on loop space.

Abstract: This paper deals rigorously with the notion of a string structure and the topological obstruction to its existence. The question of orienting loop space is discussed and shown to be directly analogous to orientation of finite dimensional manifolds. Finally, equivariant string structures are considered

Generic bifurcation of Hamiltonian vector fields with symmetry

Abstract: One of the goals of this paper is to describe explicitly the generic movement of eigenvalues through a one-to-one resonance in a linear Hamiltonian system which is equivariant with respect to a symplectic representation of a compact Lie group. We classify this movement, and hence answer the question of when the collisions are 'dangerous' in the sense of Krein by using a combination of group theory and definiteness properties of the associated quadratic Hamiltonian. For example, for systems with no symmetry or O(2) symmetry generically the eigenvalues split, whereas for systems with S1 symmetry, generically the eigenvalues may split or pass. It is in this last case that one has to use both group theory and energetics to determine the generic eigenvalue movement. The way energetics and group theory are combined is summarized in table 1. The result is to be contrasted with the bifurcation of steady states (zero eigenvalue) where one can use either group theory alone (Golubitsky and Stewart) or definiteness properties of the Hamiltonian (Cartan-Oh) to determine whether the eigenvalues split or pass on the imaginary axis.

Highly efficient estimators of multivariate location with high breakdown point

Abstract: We propose an affine equivariant estimator of multivariate location that combines a high breakdown point and a bounded influence function with high asymptotic efficiency. This proposal is basically a location $M$-estimator based on the observations obtained after scaling with an affine equivariant high breakdown covariance estimator. The resulting location estimator will inherit the breakdown point of the initial covariance estimator and within the location-covariance model only the $M$-estimator will determine the type of influence function and the asymptotic behaviour. We prove consistency and asymptotic normality and obtain the breakdown point and the influence function.

Non-self-dual Yang-Mills connections with quadrupole symmetry

Abstract: We prove the existence of non-self-dual Yang-Mills connections onSU(2) bundles over the four-sphere, specifically on all bundles with second Chern number not equal±1. We study connections equivariant under anSU(2) symmetry group to reduce the effective dimensionality from four to one, and then use variational techniques. The existence of non-self-dualSU(2) YM connections on the trivial bundle (second Chern number equals zero) has already been established by Sibner, Sibner, and Uhlenbeck via different methods.

Yang-Mills fields which are not self-dual

Abstract: The purpose of this paper is to prove the existence of a new family of non-self-dual finite-energy solutions to the Yang-Mills equations on Euclidean four-space, withSU(2) as a gauge group. The approach is that of “equivariant geometry:” attention is restricted to a special class of fields, those that satisfy a certain kind of rotational symmetry, for which it is proved that (1) a solution to the Yang-Mills equations exists among them; and (2) no solution to the self-duality equations exists among them. The first assertion is proved by an application of the direct method of the calculus of variations (existence and regularity of minimizers), and the second assertion by studying the symmetry properties of the linearized self-duality equations. The same technique yields a new family of non-self-dual solutions on the complex projective plane.

Exploiting symmetry in boundary element methods

Abstract: Linear operator equations $\mathcal {L}f = g$ are considered in the context of boundary element methods, where the operator $\mathcal {L}$ is equivariant, i.e., commutes with the actions of a given finite symmetry group. By introducing a generalization of Reynolds projectors, a decomposition of the identity operator is constructed, which in turn allows the decomposition of the problem $\mathcal {L}f = g$ into a finite number of symmetric subproblems. The data function g does not need to possess any symmetry properties. It is shown that analogous reductions are possible for discretizations. An explicit construction of the corresponding reduced system matrices is given. This effects a considerable reduction in the computational complexity. For example, in the case of the isometry group of the 3-cube, the computational complexity of a direct linear equation solver for full matrices is reduced by 99.65 percent. Specific decompositions of the identity are given for most of the significant finite isometry group...

Equivariant Euler-Poincaré characteristics and tameness

Abstract: © Société mathématique de France, 1992, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

The equivariant Hurewicz map

Abstract: Let G be a compact Lie group, Y be a based G-space, and V be a G-representation. If π V G (Y) is the equivariant homotopy group of Y in dimension V and H V G (Y) fis the equivariant ordinary homology group of Y with Burnside ring coefficients in dimension V, then there is an equivariant Hurewicz map h :π V G (Y)→H V G (Y). One should not expect this map to be an isomorphism, since H V G (Y) must be a module over the Burnside ring, but π V G (Y) need not be. However, here it is shown that, under the obvious connectivity conditions on Y, this map induces an isomorphism between H V G (Y) and an algebraically defined modification of π V G (Y)

A period mapping in universal Teichm\"uller space

Abstract: In previous work it had been shown that the remarkable homogeneous space $M= \operatorname{Diff}(S^1)/\operatorname{PSL} (2,\Bbb{R})$ sits as a complex analytic and K\"ahler submanifold of the Universal Teichm\"uller Space. There is a natural immersion $\Pi$ of $M$ into the infinite-dimensional version (due to Segal) of the Siegel space of period matrices. That map $\Pi$ is proved to be injective, equivariant, holomorphic, and K\"ahler-isometric (with respect to the canonical metrics). Regarding a period mapping as a map describing the variation of complex structure, we explain why $\Pi$ is an infinite-dimensional period mapping.

A period mapping in universal Teichmüller space

Abstract: In previous work it had been shown that the remarkable homogeneous space M=Diff(S 1 )/PSL(2, R) sits as a complex analytic and Kahler submanifold of the Universal Teichmuller Space. There is a natural immersion Π of M into the infinite-dimensional version (due to Segal) of the Siegel space of period matrices. That map Π is proved to be injective, equivariant, holomorphic, and Kahler-isometric (with respect to the canonical metrics). Regarding a period mapping as a map describing the variation of complex structure, we explain why Π is an infinite-dimensional period mapping

Isotropic geometry and twistors in higher dimensions. I. The generalized Klein correspondence and spinor flags in even dimensions

Abstract: An equivariant diffeomorphism is given between flag varieties of totally isotropic subspaces of a complex inner product space and certain orbits under the complex spin group within flag varieties of the associated spinor modules. This allows a generalization of the Klein correspondence underlying twistor theory, applicable to arbitrary (even) dimensional conformally compactified Minkowski space and its isotropic subspaces.

Bifurcation to rotating waves in equations with O (2)-symmetry

Abstract: This paper considers the detection and calculation of bifurcations from steady-state solutions to rotating (or travelling) wave solutions of the time-dependent nonlinear problem $dx/dt + g(x,\lambda ) = 0$, where g is equivariant with respect to an action of the group $O(2)$. It follows from the equivariance condition that at every nontrivial, steady-state solution $(x,\lambda ),g_x (x,\lambda )$ has a zero eigenvalue with a one-dimensional null space. It is shown that when a second real eigenvalue passes through zero as $\lambda $ varies, and the null space of $g_x (x,\lambda )$ remains one-dimensional, then bifurcation to rotating waves occurs subject to a nondegeneracy condition.It is known that rotating wave solutions satisfy a “steady-state” equation. A phase condition is added and a reflectional symmetry that is broken when bifurcation to rotating waves occurs is defined. Thus standard, steady-state, symmetry-breaking bifurcation theory can be used to analyse this type of bifurcation. An extended sy...

$G$-transversality revisited

Abstract: We express the obstruction to G-transversality in two related ways: as an element in the quotient of homotopical equivariant normal bordism by geometric bordism and also as the obstruction in a homotopy lifting problem. In both cases, our formulation is given in terms of the initial data only, in contrast to earlier theories. We also obtain related obstructions to equivariant "near" transversality, i.e., transversality using arbitrarily small homotopies.

Reduction and Equivariant Branching Lemma: Dynamical systems, evolution PDEs, and gauge theories

Abstract: We review and recast the Equivariant Branching Lemma-which has proved a remarkable tool in linearly equivariant bifurcation theory-and consider its extension to the case of nonlinear (Lie-point) symmetries. This is then applied to gauge theories and gauge theoretic problems, and to nonlinear evolution PDE's; the paper also contains an original setting of Lie-point symmetries for evolution PDEs, modelled on the dynamical systems setting.