Showing papers on "Equivariant map published in 1996"

Equivariant adaptive source separation

Abstract: Source separation consists of recovering a set of independent signals when only mixtures with unknown coefficients are observed. This paper introduces a class of adaptive algorithms for source separation that implements an adaptive version of equivariant estimation and is henceforth called equivariant adaptive separation via independence (EASI). The EASI algorithms are based on the idea of serial updating. This specific form of matrix updates systematically yields algorithms with a simple structure for both real and complex mixtures. Most importantly, the performance of an EASI algorithm does not depend on the mixing matrix. In particular, convergence rates, stability conditions, and interference rejection levels depend only on the (normalized) distributions of the source signals. Closed-form expressions of these quantities are given via an asymptotic performance analysis. The theme of equivariance is stressed throughout the paper. The source separation problem has an underlying multiplicative structure. The parameter space forms a (matrix) multiplicative group. We explore the (favorable) consequences of this fact on implementation, performance, and optimization of EASI algorithms.

Equivariant Gromov-Witten invariants

Abstract: The objective of this paper is to describe some construction and applications of the equivariant counterpart to the Gromov-Witten (GW) theory, i.e. intersection theory on spaces of (pseudo-) holomorphic curves in (almost-) Kahler manifolds. Given a Killing action of a compact Lie group G on a compact Kahler manifold X, the equivariant GW-theory provides, as we will show in Section 3, the equivariant cohomology space H G(X) with a Frobenius structure (see [2]). We discuss applications of the equivariant theory to the computation ([7],[11]) of quantum cohomology algebras of flag manifolds (Section 5), to the simultaneous diagonalization of the quantum cup-product operators (Sections 7,8), to the S-equivariant Floer homology theory on the loop space LX (see Section 6 and [10],[9]) and to a “quantum” version of the Serre duality theorem (Section 12). In Sections 9 — 11 we combine the general theory developed in Sections 1 — 6 with the fixed point localization technique [3] in order to prove the mirror conjecture (in the form suggested in [10]) for projective complete intersections. By the mirror conjecture one usually means some intriguing relations (discovered by physicists) between symplectic and complex geometry on a compact Kahler Calabi–Yau nfold and respectively complex and symplectic geometry on another Calabi-Yau n-fold called the mirror partner of the former one. The remakable application [16] of the mirror conjecture to enumeration of rational curves on Calabi–Yau 3-folds (1991, see the theorem below) raised a number of new mathematical problems — challenging maturity tests for modern methods of symplectic topology.

Equivariant Gromov - Witten Invariants

Abstract: We develop general theory of equivariant quantum cohomology for ample Kahler manifolds and prove the mirror conjecture for projective complete intersections.

Equivariant intersection theory

Abstract: In this paper we develop an equivariant intersection theory for actions of algebraic groups on algebraic schemes. The theory is based on our construction of equivariant Chow groups. They are algebraic analogues of equivariant cohomology groups which satsify the formal properties of ordinary Chow groups. In addition, they enjoy many of the properties of equivariant cohomology. The principal results are: (1) We prove the existence of canonical intersection products on the Chow groups of geometric quotients of smooth varieties- even when the stabilizers of geometric points are non-reduced. (2) We construct a Todd class map from equivariant $K$-theory of coherent sheaves to a completion of equivariant Chow groups, and prove that a completion of equivariant $K$-theory is isomorphic to the completion of equivariant Chow groups. (3) We prove a localization theorem for torus actions and use it to give a characteristic free proof of the Bott residue formula for actions of tori on complete smooth varieties.

Automorphisms, root systems, and compactifications of homogeneous varieties

Abstract: Let G be a complex semisimple group and let H C G be the group of fixed points of an involutive automorphism of G. Then X = G/H is called a symmetric variety. In [CP], De Concini and Procesi have constructed an equivariant compactification X which has a number of remarkable properties, some of them being: i) The boundary is the union of divisors D1,... , Dr. ii) There are exactly 2r orbits. Their closures are the intersections Di1 n .nDi (even schematically). In particular, there is only one closed orbit. iii) In case G is of adjoint type, all orbit closures are smooth. It is called the wonderful embedding of X or a complete symmetric variety and is the foundation for most deeper results about X. Independently, Luna and Vust developed in [LV] a general theory of equivariant compactifications of homogeneous varieties under a connected reductive group G. In particular, they realized the reason which makes symmetric varieties behave so nicely: A Borel subgroup B has an open dense orbit in G/H. Varieties with this property are called spherical. Luna and Vust were able to describe all equivariant compactifications of them in terms of combinatorial data, very similar to torus embeddings which are actually a special case. They obtained in particular that every spherical embedding has only finitely many orbits. Nevertheless, the reason for the existence of a compactification with properties i)-iii) remained mysterious. Then Brion and Pauer established a relation with the automorphism group. They proved in [BP]: A spherical variety X = G/H possesses an equivariant compactification with exactly one closed orbit if and only if AutG X = NG (H)/H is finite. In this case there is a unique one which dominates all others: the wonderful compactification X. They also showed that the orbits of X correspond to the faces of a strictly convex polyhedral cone Z. Then properties i) and ii) above are equivalent to Z being simplicial. This fact is much deeper and was proved by Brion in [Brl]. In fact he showed much more. Let F be the set of characters of B which are the characters of a rational B-eigenfunction on X. This is a finitely generated free abelian group. Then the cone Z is a subset of the real vector space Hom(F, R). Brion showed that there is a finite reflection group Wx acting on F such that Z is one of its Weyl chambers. In case of a symmetric variety, Wx is its little Weyl group.

Limit distributions of expanding translates of certain orbits on homogeneous spaces

Abstract: LetL be a Lie group and λ a lattice inL. SupposeG is a non-compact simple Lie group realized as a Lie subgroup ofL and $$\overline {GA} = L$$ . LetaeG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G)). Consider the expanding horospherical subgroup ofG associated toa defined as U+ ={geG:a −n gan} →e as n → ∞. Let Ω be a non-empty open subset ofU + andn i → ∞ be any sequence. It is showed that $$\overline { \cup _{i = 1}^\infty a^n \Omega \Lambda } = L$$ . A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeG-equivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.

The Chern character of a transversally elliptic symbol and the equivariant index

Abstract: Let G be a compact Lie group acting on a compact manifold M. In this article, we associate to a G-transversally elliptic symbol on M a G- invariant generalized function on G, constructed in terms of equivariant closed dierential forms on the cotangent bundle TM.

On a transformation and re-transformation technique for constructing an affine equivariant multivariate median

Abstract: An affine equivariant version of multivariate median is introduced. The proposed median is easy to compute and has some appealing geometric features that are related to the configuration of a multivariate data cloud. The transformation and re-transformation approach used in the construction of the median has some fundamental connection with the data driven co-ordinate system considered by Chaudhuri and Sengupta (1993, Journal of the American Statistical Association). Large sample statistical properties of the median are discussed and finite sample performance is investigated using Monte Carlo simulations.

Equivariant Localization of Path Integrals

Abstract: We review equivariant localization techniques for the evaluation of Feynman path integrals. We develop systematic geometric methods for studying the semi-classical properties of phase space path integrals for dynamical systems, emphasizing the relations with integrable and topological quantum field theories. Beginning with a detailed review of the relevant mathematical background -- equivariant cohomology and the Duistermaat-Heckman theorem, we demonstrate how the localization ideas are related to classical integrability and how they can be formally extended to derive explicit localization formulas for path integrals in special instances using BRST quantization techniques. Various loop space localizations are presented and related to notions in quantum integrability and topological field theory. We emphasize the common symmetries that such localizable models always possess and use these symmetries to discuss the range of applicability of the localization formulas. A number of physical and mathematical applications are presented in connection with elementary quantum mechanics, Morse theory, index theorems, character formulas for semi-simple Lie groups, quantization of spin systems, unitary integrations in matrix models, modular invariants of Riemann surfaces, supersymmetric quantum field theories, two-dimensional Yang-Mills theory, conformal field theory, cohomological field theories and the loop expansion in quantum field theory. Some modern techniques of path integral quantization, such as coherent state methods, are also discussed. The relations between equivariant localization and other ideas in topological field theory, such as the Batalin-Fradkin-Vilkovisky and Mathai-Quillen formalisms, are presented.

On equivariant quantum cohomology

Abstract: Using the Kontsevich's moduli space of stable maps, we define the equivariant quantum cohomology for generalized flag varieties and make a rigorous computation of quantum cohomology of flag varieties.

On Poisson actions of compact Lie groups on symplectic manifolds

Abstract: Let G(P) be a compact simple Poisson-Lie group equipped with a Poisson structure P, and (M, omega) be a symplectic manifold. Assume that M carries a Poisson action of G(P), and there is an equivariant moment map in the sense of Lu and Weinstein which maps

Equivariant index formulas for orbifolds

Abstract: Let P be a smooth manifold. Let H be a compact Lie group acting on P . We assume that the action of H is infinitesimally free, that is the stabilizer H(y) of any point y ∈ P is a finite subgroup of H. We write the action of H on the right. The quotient space P/H is an orbifold (if H acts freely, then P/H is a manifold). Reciprocally any orbifold M can be presented this way: for example, one might choose P to be the bundle of orthonormal frames for a choice of a metric on M and H = O(n) if n = dimM . We will assume that there is a compact Lie group G acting on P such that its action commutes with the action of H. We will write the action of G on the left. Then the space P/H is provided with a G-action. Such data (P,H,G) will be our definition of a presented G-orbifold. We will say shortly that P/H is a G-orbifold. Consider a compact G-orbifold P/H. A tangent vector on P tangent at y ∈ P to the orbit H · y will be called a vertical tangent vector. Let T ∗ HP be the subbundle of T P orthogonal to all vertical vectors. We will say that T ∗ HP is the horizontal cotangent space. We denote by (y, ξ) a point in T P . Consider two (G×H)-equivariant vector bundles E on P . Let Γ(P, E) be the spaces of smooth sections of E. Let

On realizing measured foliations via quadratic differentials of harmonic maps toR-trees

Abstract: We give a brief, elementary and analytic proof of the theorem of Hubbard and Masur [HM] (see also [K], [G]) that every class of measured foliations on a compact Riemann surfaceR of genusg can be uniquely represented by the vertical measured foliation of a holomorphic quadratic differential onR. The theorem of Thurston [Th] that the space of classes of projective measured foliations is a 6g—7 dimensional sphere follows immediately by Riemann-Roch. Our argument involves relating each representative of a class of measured foliations to an equivariant map from \( \tilde R \) to anR-tree, and then finding an energy minimizing such map by the direct method in the calculus of variations. The normalized Hopf differential of this harmonic map is then the desired differential.

Zonoid Data Depth: Theory and Computation

Abstract: A new notion of data depth in d-space is presented, called the zonoid data depth. It is affine equivariant and has useful continuity and monotonicity properties. An efficient algorithm is developed that calculates the depth of a given point with respect to a d-variate empirical distribution.

On normality of the closure of a generic torus orbit in G∕P

Abstract: In this paper we consider generic orbits for the action of a maximal torus T in a connected semisimple algebraic group G on the generalized flag variety G/P, where P is a parabolic subgroup of G containing T. The union of all generic T-orbits is an open dense (possibly proper, if P is not a Borel subgroup) subset of the intersection of the big cells in G/P. We prove that the closure of a generic Γ-orbit in G/P is a normal equivariant T-embedding (whose fan we explicitely describe). Moreover, the closures of any two generic Γ-orbits are isomorphic as equivariant T-embeddings.

Momentum mappings and poisson cohomology

Abstract: We analyze the question of existence and uniqueness of equivariant momentum mappings for Poisson actions of Poisson Lie groups. A necessary and sufficient condition for the equivariant momentum mapping to be unique is given. The existence problem is solved under some extra hypotheses, for example, when the action preserves the Poisson structure. In this case, the problem is closely related to the triviality of the induced group action on the Poisson cohomology. This action is shown to be trivial whenever the group is compact or semisimple. Conceptually, these results rely upon a version of “Poisson calculus” developed here to make one-forms on a Poisson manifold induce a “flow” preserving the Poisson structure. In the general case, obstructions to the existence of an infinitesimal version of an equivariant momentum mapping are found. Using Lie algebra cohomology with coefficients in Frechet modules, we show that the obstructions vanish, and the infinitesimal mapping exists, when the group is compact semisimple. We also prove the rigidity of compact group actions preserving the Poisson structure on a compact manifold and calculate the Poisson cohomology of the Poisson homogeneous space .

_{∞}-ring structures for Tate spectra

Abstract: Let G be a compact Lie group and kG a G spectrum (as defined in [3, Section I.2]). Greenlees and May ([2]) have defined an associated G-spectrum t(kG) called the Tate spectrum. They observe that if kG is a ring G-spectrum then there is an induced ring G-spectrum structure on t(kG), and that if kG is homotopy-commutative then t(kG) will also be homotopycommutative (see [2, Proposition 3.5]). It is therefore natural to ask whether an equivariant E∞-ring structure on kG induces an equivariant E∞-ring structure on t(kG) (we will recall the definition in a moment). We offer both positive and negative answers to this question. On the positive side, we show that t(kG) inherits a structure which is somewhat weaker than an equivariant E∞-ring structure, but which should be adequate for most practical purposes. To explain this, let us recall from [3, Example VII.1.4] that to each G-universe U is associated an equivariant operad L(U). Let us fix a complete G universe U and let V denote the trivial G-universe U. An equivariant E∞-ring structure is defined to be an action of an equivariant operad equivalent to L(U) (see [3, Definitions VII.2.1 and VII.1.2 and Remark VII.1.3]). Let us define an E′ ∞-ring structure to be an action of an equivariant operad equivalent to L(V ); since G acts trivially on L(V ) we can rephrase this by saying that an E′ ∞-ring structure is an action of a nonequivariant E∞ operad through G-maps. Since there is a map of operads L(V ) → L(U), an equivariant E∞ structure specializes to an E′ ∞ structure. On the other hand, Remark VII.2.5 of [3] shows that if kG is an E ′ ∞-ring spectrum then the fixed point spectra (kG) H have (nonequivariant) E∞-ring structures which are consistent as H varies; this is likely to be the point most relevant for applications. Our positive result is:

Relative Geometric Invariant Theory and Universal Moduli Spaces

Abstract: We expose in detail the principle that the relative geometric invariant theory of equivariant morphisms is related to the GIT for linearizations near the boundary of the G-effective ample cone. We then apply this principle to construct and reconstruct various universal moduli spaces. In particular, we constructed the universal moduli space over of Simpson’s p-semistable coherent sheaves and a canonical rational morphism from the universal Hilbert scheme over to a compactified universal Picard.

Topological analysis of chaos in equivariant electronic circuits

Abstract: Chaotic oscillations in an electronic circuit are studied by recording two time series simultaneously. The chaotic dynamics is characterized by using topological analysis. A comparison with two models is also discussed. Some prescriptions are given in order to take into account the symmetry properties of the experimental system to perform the topological analysis.

Evolution of a multimodal map induced by an equivariant vector field

Abstract: It has been shown that the topological characterization of an equivariant system should preferably be achieved by working in a fundamental domain generated by the symmetry properties appearing in the phase space. In this paper, we discuss the case when the equivariance of the studied system is taken into account to study the evolution of the population of periodic orbits when a control parameter is varied. The Burke - Shaw system is considered here as an example. It is shown that the equivariance of this system may be used to reduce the multimodal first-return map in a Poincare section to a unimodal map. A relationship between four-symbol sequences and two-symbol sequences is given. The non-trivial evolution of the orbit spectrum of a multimodal map is then predicted from the much simpler unimodal map to which the multimodal map reduces.

Equivariant homotopy and cohomology theory : dedicated to the memory of Robert J. Piacenza

Abstract: Introduction Equivariant cellular and homology theory Postnikov systems, localization, and completion Equivariant rational homotopy theory Smith theory Categorical constructions equivariant applications The homotopy theory of diagrams Equivariant bundle theory and classifying spaces The Sullivan conjecture An introduction to equivariant stable homotopy $G$-CW$(V)$ complexes and $RO(G)$-graded cohomology The equivariant Hurewicz and suspension theorems The equivariant stable homotopy category $RO(G)$-graded homology and cohomology theories An introduction to equivariant $K$-theory An introduction to equivariant cobordism Spectra and $G$-spectra change of groups duality The Burnside ring Transfer maps in equivariant bundle theory Stable homotopy and Mackey functors The Segal conjecture Generalized Tate cohomology Twisted half-smash products and function spectra Brave new algebra Brave new equivariant foundations Brave new equivariant algebra Localization and completion in complex bordism A completion theorem in complex cobordism Calculations in complex equivariant bordism Bibliography Index.