Showing papers on "Equivariant map published in 1997"

Equivariant cohomology, Koszul duality, and the localization theorem

Abstract: (1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality. This enables one to translate facts about equivariant cohomology into facts about its ordinary cohomology, and back. (3) Equivariant derived category. Many of the results in this paper apply not only to equivariant cohomology, but also to equivariant intersection cohomology. The equivariant derived category provides a framework in both of these may be considered simultaneously, as examples of ``equivariant sheaves''. We treat singular spaces on an equal footing with nonsingular ones. Along the way, we give a description of equivariant homology and equivariant intersection homology in terms of equivariant geometric cycles. Most of the themes in this paper have been considered by other authors in some context. In Sect. 1.7 we sketch the precursors that we know about. For most of the constructions in this paper, we consider an action of a compact connected Lie group K on a space X , however for the purposes of the introduction we will take K ˆ …S1† to be a torus. Invent. math. 131, 25±83 (1998)

Modular invariance of trace functions in orbifold theory

Abstract: The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise numbers of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlations functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representations of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway-Norton-Queen and to equivariant elliptic cohomology.

Equivariant Chow groups for torus actions

Abstract: We study Edidin and Graham's equivariant Chow groups in the case of torus actions. Our main results are: (i) a presentation of equivariant Chow groups in terms of invariant cycles, which shows how to recover usual Chow groups from equivariant ones; (ii) a precise form of the localization theorem for torus actions on projective, nonsingular varieties; (iii) a construction of equivariant multiplicities, as functionals on equivariant Chow groups; (iv) a construction of the action of operators of divided differences on theT-equivariant Chow group of any scheme with an action of a reductive group with maximal torusT. We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations. In particular, we obtain a presentation of the Chow ring of any smooth, projective spherical variety.

Zonoid trimming for multivariate distributions

Abstract: A family of trimmed regions is introduced for a probability distribution in Euclidean d-space. The regions decrease with their parameter $\alpha$, from the closed convex hull of support (at $\alpha = 0$) to the expectation vector (at $\alpha = 1$). The family determines the underlying distribution uniquely. For every $\alpha$ the region is affine equivariant and continuous with respect to weak convergence of distributions. The behavior under mixture and dilation is studied. A new concept of data depth is introduced and investigated. Finally, a trimming transform is constructed that injectively maps a given distribution to a distribution having a unique median.

Polygon spaces and grassmannians

Abstract: We study the moduli spaces of polygons in R^2 and R^3, identifying them with subquotients of 2-Grassmannians using a symplectic version of the Gel'fand-MacPherson correspondence. We show that the bending flows defined by Kapovich-Millson arise as a reduction of the Gel'fand-Cetlin system on the Grassmannian, and with these determine the pentagon and hexagon spaces up to equivariant symplectomorphism. Other than invocation of Delzant's theorem, our proofs are purely polygon-theoretic in nature.

An equivariant Riemann-Roch theorem for complete, simplicial toric varieties.

Abstract: Introduction. The theory of toric varieties establishes a now classical connection between algebraic geometry and convex polytopes. In particular, äs observed by Danilov in the seventies, finding a closed formula for the Todd class of complete toric varieties would have important consequences for enumeration of lattice points in convex lattice polytopes. Since then, a number of such formulas have been proposed; see [M], [Pl], [P2] The Todd class of complete simplicial toric varieties is computed in [G-G-K], using the Riemann-Roch formula of T. Kawasaki [Ka].

Theory of Degrees with Applications to Bifurcations and Differential Equations

Abstract: Elements of Differential Topology. Degree in Finite--Dimensional Spaces. Leray--Schauder Degree for Compact Fields. Nussbaum--Sadovskii Degree for Condensing Fields. Applications to Bifurcation Theory. S 1 --Equivariant Degree. Global Hopf Bifurcation Theory. Equivariant Degree of Dold--Ulrich. References. Index.

Equivariant resolution of singularities in characteristic 0

Abstract: A new proof of equivariant resolution of singularities under a finite group action in characteristic 0 is provided. We assume we know how to resolve singularities without group action. We first prove equivariant resolution of toroidal singularities. Then we reduce the general case to the toroidal case.

Localization and completion theorems for mu-module spectra

Abstract: Let G be a finite extension of a torus. Working with highly structured ring and module spectra, let M be any module over MU ; examples include all of the standard homotopical MU -modules, such as the Brown-Peterson and Morava K-theory spectra. We shall prove localization and completion theorems for the computation of M∗(BG) and M∗(BG). The G-spectrum MUG that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum SG, and there is an MUG-module MG whose underlying MU -module is M . This allows the use of topological analogues of constructions in commutative algebra. The computation ofM∗(BG) andM∗(BG) is expressed in terms of spectral sequences whose respective E2 terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring MU ∗ and its module M ∗ . The central feature of the proof is a new norm map in equivariant stable homotopy theory, the construction of which involves the new concept of a global I∗-functor with smash product.

Degree for Gradient Equivariant Maps and Equivariant Conley Index

Abstract: In order to establish some notation and terminology we recall that if f : ℝ n → ℝ n is continuous and Ω is an open bounded subset of ℝ n such that f is different from 0 on the boundary of Ω, then there is defined an integer Deg(f, Ω) — the Brouwer (or topological) degree of f with respect to Ω. Obviously, if in the place of all continuos maps and all open bounded subsets of ℝ n we take a smaller class of maps and/or a smaller class of subsets then we may try to define a topological invariant finer then the topological degree.

Noncompact drift for relative equilibria and relative periodic orbits

Abstract: In the context of equivariant dynamical systems with a compact Lie group, Gamma, of symmetries, Field and Krupa have given sharp upper bounds on the drifts associated with relative equilibria and relative periodic orbits. For relative equilibria consisting of points of trivial isotropy, the drifts correspond to tori in Gamma. Generically, these are maximal tori. Analogous results hold when there is a nontrivial isotropy subgroup Sigma, with Gamma replaced by N(Sigma)/Sigma. In this paper, we generalize the results of Field and Krupa to noncompact Lie groups. The drifts now correspond to tori or lines (unbounded copies of R) in Gamma and generically these are maximal tori or lines. Which of these drifts is preferred, compact or unbounded, depends on Gamma: there are examples where compact drift is preferred (Euclidean group in the plane), where unbounded drift is preferred (Euclidean group in three-dimensional space) and where neither is preferred (Lorentz group). Our results partially explain the quasiperiodic (Winfree) and linear (Barkley) meandering of spirals in the plane, as well as the drifting behaviour of spiral bound pairs (Ermakova et al). In addition, we obtain predictions for the drifting of the scroll solutions (scroll waves and scroll rings, twisted and linked) considered by Winfree and Strogatz.

Symmetry-breaking bifurcations on cubic lattices

Abstract: Steady-state symmetry-breaking bifurcations on the simple (SC), face-centred (FCC) and body-centred (BCC) cubic lattices are considered, corresponding to the 6-, 8- and 12-dimensional representations of the group . Methods of equivariant bifurcation theory are used to identify all primary solution branches and to determine their stability; branches with submaximal isotropy are generic for both the FCC and BCC lattices. Complete analysis of the local branching behaviour in the SC (three primary branches) and FCC (five primary branches) cases is given. The BCC case is substantially more complex, owing to the presence of a quadratic equivariant. A degeneracy that arises in the presence of an additional reflection symmetry is analysed first using a normal form truncated at third order. This problem, in which no quadratic equivariants are present, yields 10 primary branches with maximal isotropy and five with submaximal isotropy. The unfolding of the degeneracy is used to show that seven primary branches (six maximal and one submaximal) persist in the generic case, and to determine the form and properties of secondary branches. In certain cases higher-order terms are necessary. The study is motivated by the problem of pattern formation in three spatial dimensions, and extends earlier work by De Wit and co-workers.

Timely Communication: Symmetry and the Karhunen--Loève Analysis

Abstract: The Karhunen--Loeve (K--L) analysis is widely used to generate low-dimensional dynamical systems, which have the same low-dimensional attractors as some large-scale simulations of PDEs. If the PDE is symmetric with respect to a symmetry group G, the dynamical system has to be equivariant under G to capture the full phase space. It is shown that symmetrizing the K--L eigenmodes instead of symmetrizing the data leads to considerable computational savings if the K-L analysis is done in the snapshot method. The feasibility of the approach is demonstrated with an analysis of Kolmogorov flow.

Localization of virtual classes

Abstract: We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.

Twisted N=2 supersymmetry with central charge and equivariant cohomology

Abstract: We present an equivariant extension of the Thom form with respect to a vector field action, in the framework of the Mathai-Quillen formalism. The associated Topological Quantum Field Theories correspond to twisted N=2 supersymmetric theories with a central charge. We analyze in detail two different cases: topological sigma models and non-abelian monopoles on four-manifolds.

Equivariant holomorphic Morse inequalities. I. Heat kernel proof

Abstract: Assume that the circle group acts holomorphically on a compact K\\\"ahler manifold with isolated fixed points and that the action can be lifted holomorphically to a holomorphic Hermitian vector bundle. We give a heat kernel proof of the equivariant holomorphic Morse inequalities. We use some techniques developed by Bismut and Lebeau. These inequalities, first obtained by Witten using a different argument, produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomologies in terms of the data of the fixed points.

On the Stability of Stationary Wave Maps

Abstract: Equivariant wave maps from \(\) into \(\) have smooth, stationary solutions which are critical points of the energy subject to constant charge. These solutions are globally stable under equivariant perturbations. Consequently, there exists a large set of initial data, with no degree or energy restrictions, for which the Cauchy problem is globally well-posed.

Equivariant Reidemeister torsion on symmetric spaces

Abstract: We calculate explicitly the equivariant Ray-Singer torsion for all symmetric spaces \(G/K\) of compact type with respect to the action of \(G\). We show that it equals zero except for the odd-dimensional Grasmannians and the space \(\vec{SU}(3)/\vec{SO}(3)\). As a corollary, we classify up to diffeomorphism all isometries of these spaces which are homotopic to the identity; also, we classify their quotients by finite group actions up to homeomorphism.

Equivariant Holomorphic Morse Inequalities III: Non-Isolated Fixed Points

Abstract: We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kahler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kahler manifold in terms of those of the fixed-point set. We apply the inequalities to obtain relations of Hodge numbers of the connected components of the fixed-point set and the whole manifold. We also investigate the consequences in geometric quantization, especially in the context of symplectic cutting.

Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity

Abstract: An asymptotically linear Hamiltonian system with strong resonance at infinity is considered. The existence of multiple periodic solutions is proved via variational methods in an equivariant setting.