Showing papers on "Equivariant map published in 2003"

Multi-instanton calculus and equivariant cohomology

Abstract: We present a systematic derivation of multi-instanton amplitudes in terms of ADHM equivariant cohomology. The results rely on a supersymmetric formulation of the localization formula for equivariant forms. We examine the cases of N = 4 and N = 2 gauge theories with adjoint and fundamental matter.

Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks

Abstract: A coupled cell system is a network of dynamical systems, or "cells," coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only mechanism that can create such states in a coupled cell system and show that it is not. The key idea is to replace the symmetry group by the symmetry groupoid, which encodes in- formation about the input sets of cells. (The input set of a cell consists of that cell and all cells connected to that cell.) The admissible vector fields for a given graph—the dynamical systems with the corresponding internal dynamics and couplings—are precisely those that are equivariant under the symmetry groupoid. A pattern of synchrony is "robust" if it arises for all admissible vector fields. The first main result shows that robust patterns of synchrony (invariance of "polydiagonal" subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an equivalence relation on cells is "balanced." The second main result shows that admissible vector fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled cell network, the "quotient network." The existence of quotient networks has surprising implications for synchronous dynamics in coupled cell systems.

Puzzles and (equivariant) cohomology of Grassmannians

Abstract: The product of two Schubert cohomology classes on a Grassmannian ${\rm Gr}_k (\mathbb{c}^n)$ has long been known to be a positive combination of other Schubert classes, and many manifestly positive formulae are now available for computing such a product (eg, the Littlewood-Richardson rule or the more symmetric puzzle rule from A Knutson, T Tao, and C Woodward [KTW]) Recently, W~Graham showed in [G], nonconstructively, that a similar positivity statement holds for {\em $T$-equivariant} cohomology (where the coefficients are polynomials) We give the first manifestly positive formula for these coefficients in terms of puzzles using an ``equivariant puzzle piece'' The proof of the formula is mostly combinatorial but requires no prior combinatorics and only a modicum of equivariant cohomology (which we include) As a by-product the argument gives a new proof of the puzzle (or Littlewood-Richardson) rule in the ordinary-cohomology case, but this proof requires the equivariant generalization in an essential way, as it inducts backwards from the ``most equivariant'' case This formula is closely related to the one in A Molev and B Sagan [MS] for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [G] We include a cohomological interpretation of their problem and a puzzle formulation for it

Topology of Singular Spaces and Constructible Sheaves

Abstract: 1 Thom-Sebastiani Theorem for constructible sheaves.- 1.1 Milnor fibration.- 1.1.1 Cohomological version of a Milnor fibration.- 1.1.2 Examples.- 1.2 Thom-Sebastiani Theorem.- 1.2.1 Preliminaries and Thom-Sebastiani for additive functions.- 1.2.2 Thom-Sebastiani Theorem for sheaves.- 1.3 The Thom-Sebastiani Isomorphism in the derived category.- 1.4 Appendix: Kunneth formula.- 2 Constructible sheaves in geometric categories.- 2.0.1 The basic results.- 2.0.2 Definable spaces.- 2.1 Geometric categories.- 2.2 Constructible sheaves.- 2.3 Constructible functions.- 3 Localization results for equivariant constructible sheaves.- 3.1 Equivariant sheaves.- 3.1.1 Equivariant sheaves and monodromic complexes.- 3.1.2 Equivariant derived categories.- 3.1.3 Examples and stalk formulae.- 3.2 Localization results for additive functions.- 3.3 Localization results for Grothendieck groups and trace formulae.- 3.3.1 Grothendieck groups.- 3.3.2 Trace formulae.- 3.4 Equivariant cohomology.- 4 Stratification theory and constructible sheaves.- 4.1 Stratification theory.- 4.1.1 A cohomological version of the first isotopy lemma.- 4.1.2 Comparison of different regularity conditions.- 4.1.3 Micro-local characterization of constructible sheaves.- 4.2 Constructible sheaves on stratified spaces.- 4.2.1 Cohomologically cone-like stratifications.- 4.2.2 Stability results for constructible sheaves.- 4.3 Base change properties.- 4.3.1 Some constructions for stratifications.- 4.3.2 Base change isomorphisms.- 5 Morse theory for constructible sheaves.- 5.0.1 Real stratified Morse theory.- 5.0.2 Complex stratified Morse theory.- 5.0.3 Introduction to characteristic cycles.- 5.1 Stratified Morse theory, part I.- 5.1.1 Local Morse data.- 5.1.2 Normal Morse data.- 5.1.3 Morse theory for a stratified space with corners.- 5.2 Characteristic cycles and index formulae.- 5.2.1 Index formulae and Euler obstruction.- 5.2.2 A specialization argument.- 5.3 Stratified Morse theory, part II.- 5.3.1 Normal Morse data are independent of choices.- 5.3.2 Splitting of the local Morse data.- 5.3.3 Normal Morse data and micro-localization.- 5.4 Vanishing cycles.- 6 Vanishing theorems for constructible sheaves.- Introduction: Results and examples.- 6.0.1 (Co)stalk properties.- 6.0.2 Intersection (co)homology and perverse sheaves.- 6.0.3 Vanishing results in the complex context.- 6.0.4 Nearby and vanishing cycles.- 6.0.5 Artin-Grothendieck type theorems.- 6.0.6 Applications to constructible functions.- 6.1 Proof of the results.

Twisted Orbifold K-Theory

Abstract: We use equivariant methods to define and study the orbifold K-theory of an orbifold X. Adapting techniques from equivariant K-theory, we construct a Chern character and exhibit a multiplicative decomposition for K*orb(X)⊗ℚ, in particular showing that it is additively isomorphic to the orbifold cohomology of X. A number of examples are provided. We then use the theory of projective representations to define the notion of twisted orbifold K–theory in the presence of discrete torsion. An explicit expression for this is obtained in the case of a global quotient.

Equivariant wave maps in two space dimensions

Abstract: Singularities of corotational wave maps from (1 + 2)-dimensional Minkowski space into a surface N of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 2 into ℕ. In consequence, for noncompact target surfaces of revolution, the Cauchy problem for wave maps is globally well-posed. © 2003 Wiley Periodicals, Inc.

On the Equivariant Tamagawa number conjecture for Tate motives

Abstract: Let L be a finite abelian extension of ℚ and let K be any subfield of L. For each integer r with r≤0 we prove the Equivariant Tamagawa Number Conjecture for the pair (h 0(Spec(L))(r),ℤ[½][Gal(L/K)]).

Framed Discs Operads and Batalin–Vilkovisky Algebras

Abstract: The framed n-discs operad fD(n) is studied as semidirect product of SO(n) and the little n-discs operad. Our equivariant recognition principle says that a grouplike space acted on by fD(n) is equivalent to the n-fold loop space on an SO(n)-space. Examples of fD(2)-spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of fD(n), which produces higher Batalin-Vilkovisky algebra structures for n even. We study quadratic duality for semidirect product operads and compute the double loop space homology of a manifold as BV-algebra.

Equivariant spectral triples on the quantum SU(2) group

Abstract: We characterize all equivariant odd spectral triples for the quantum SU(2) group acting on its L2-space and having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and given any element of the K-homology group of SUq(2), there is an equivariant odd spectral triple of dimension 3 inducing that element. The method employed to get equivariant spectral triples in the quantum case is then used for classical SU(2), and we prove that for p<4, there does not exist any equivariant spectral triple with nontrivial K-homology class and dimension p acting on the L2-space.

IRREDUCIBLE AUTOMORPHISMS OF $F_{n}$ HAVE NORTH–SOUTH DYNAMICS ON COMPACTIFIED OUTER SPACE

Abstract: We show that if an automorphism of a non-abelian free group $F_n$ is irreducible with irreducible powers, it acts on the boundary of Culler–Vogtmann’s outer space with north–south dynamics: there are two fixed points, one attracting and one repelling, and orbits accumulate only on these points. The main new tool we use is the equivariant assignment of a point $Q(X)$ to any end $X\in\partial F_n$ , given an action of $F_n$ on an $\bm{R}$ -tree $T$ with trivial arc stabilizers; this point $Q(X)$ may be in $T$ , or in its metric completion, or in its boundary. AMS 2000 Mathematics subject classification: Primary 20F65; 20E05; 20E08

Proper Group Actions and the Baum-Connes Conjecture

Abstract: Equivariant K-Homology of the Classifying Space for Proper Actions.- On the Baum-Connes Assembly Map for Discrete Groups.

Quantum Groups, the loop Grassmannian, and the Springer resolution

Abstract: We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the loop Grassmannian for the Langlands dual group. The equivalence (1)-(2) is an `enhancement' of the known expression for quantum group cohomology in terms of nilpotent variety, due to Ginzburg-Kumar. The equivalence (2)-(3) is a step towards resolving an old mystery surrounding the existense of two completely different realizations of the affine Hecke algebra which have played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a p-adic reductive group, while the other is in terms of equivariant K-theory of a complex (Steinberg) variety for the dual group. Our equivalence (2)-(3) may be viewed as a `categorification' of the isomorphism between the corresponding two geometric realizations of the fundamental polynomial representation of the affine Hecke algebra. The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on representation theory of Kac-Moody algebras). It also gives way to proving Humphreys' conjectures on tilting U_q(g)-modules, as will be explained in a separate paper.

Steenrod operations in chow theory

Abstract: An action of the Steenrod algebra is constructed on the mod p Chow theory of varieties over a field of characteristic different from p, answering a question posed in Fulton's Intersection Theory. The action agrees with the action of the Steenrod algebra used by Voevodsky in his proof of the Milnor conjecture. However, the construction uses only basic functorial properties of equivariant intersection theory.

Invariant Euler–Lagrange Equations and the Invariant Variational Bicomplex

Abstract: In this paper, we derive an explicit group-invariant formula for the Euler- Lagrange equations associated with an invariant variational problem. The method relies on a group-invariant version of the variational bicomplex induced by a general equivariant moving frame construction, and is of independent interest.

Seiberg{Witten{Floer stable homotopy type of three-manifolds with b1 =0

Abstract: Using Furuta’s idea of nite dimensional approximation in Seiberg{Witten theory, we rene Seiberg{Witten Floer homology to obtain an invariant of homology 3{spheres which lives in the S 1 {equivariant graded suspension category. In particular, this gives a construction of Seiberg{Witten Floer homology that avoids the delicate transversality problems in the standard approach. We also dene a relative invariant of four-manifolds with boundary which generalizes the Bauer{Furuta stable homotopy invariant of closed four-manifolds.

Higher algebraic K-theory for actions of diagonalizable groups

Abstract: We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type.

Classification of Weil-Petersson isometries

Abstract: This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmuller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the Thurston classification. The second result concerns the asymptotic behavior of these geodesics. It is shown that geodesics that are equivariant with respect to independent pseudo-Anosov's diverge. It follows that subgroups of the mapping class group which contain independent pseudo-Anosov's act in a reductive manner with respect to the Weil-Petersson geometry. This implies an existence theorem for equivariant harmonic maps to the metric completion.

Theory of integer equivariant estimation with application to GNSS

Abstract: Carrier phase ambiguity resolution is the key to high-precision global navigation satellite system (GNSS) positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo. The so-called ‘fixed’ baseline estimator is known to be superior to its ‘float’ counterpart in the sense that its probability of being close to the unknown but true baseline is larger than that of the ‘float’ baseline, provided that the ambiguity success rate is sufficiently close to its maximum value of one. Although this is a strong result, the necessary condition on the success rate does not make it hold for all measurement scenarios. It is discussed whether or not it is possible to take advantage of the integer nature of the ambiguities so as to come up with a baseline estimator that is always superior to both its ‘float’ and its ‘fixed’ counterparts. It is shown that this is indeed possible, be it that the result comes at the price of having to use a weaker performance criterion. The main result of this work is a Gauss–Markov-like theorem which introduces a new minimum variance unbiased estimator that is always superior to the well-known best linear unbiased (BLU) estimator of the Gauss–Markov theorem. This result is made possible by introducing a new class of estimators. This class of integer equivariant estimators obeys the integer remove–restore principle and is shown to be larger than the class of integer estimators as well as larger than the class of linear unbiased estimators. The minimum variance unbiased estimator within this larger class is referred to as the best integer equivariant (BIE) estimator. The theory presented applies to any model of observation equations having both integer and real-valued parameters, as well as for any probability density function the data might have.

Equivariant K-theory and equivariant cohomology

Abstract: For T an abelian compact Lie group, we give a description of T-equivariant K-theory with complex coefficients in terms of equivariant cohomology. In the appendix we give applications of this by extending results of Chang-Skjelbred and Goresky-Kottwitz-MacPherson from equivariant cohomology to equivariant K-theory.

Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone

Abstract: In math.AG/0005152 a certain $t$-structure on the derived category of equivariant coherent sheaves on the nil-cone of a simple complex algebraic group was introduced (the so-called perverse $t$-structure corresponding to the middle perversity). In the present note we show that the same $t$-structure can be obtained from a natural quasi-exceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig and Vogan (and obtained by other means in math.RT/0010089).

Pattern-equivariant functions and cohomology

Abstract: The cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and, therefore, more easily accessible. It is based on generalizing the notion of equivariance from lattices to point patterns of finite local complexity.

Equivariant Degree Theory

Abstract: This book presents a new degree theory for maps which commute with a group of symmetries. This degree is no longer a single integer but an element of the group of equivariant homotopy classes of maps between two spheres and depends on the orbit types of the spaces. The authors develop completely the theory and applications of this degree in a self-contained presentation starting with only elementary facts. The first chapter explains the basic tools of representation theory, homotopy theory and differential equations needed in the text. Then the degree is defined and its main abstract properties are derived. The next part is devoted to the study of equivariant homotopy groups of spheres and to the classification of equivariant maps in the case of abelian actions. These groups are explicitely computed and the effects of symmetry breaking, products and composition are thorougly studied. The last part deals with computations of the equivariant index of an isolated orbit and of an isolated loop of stationary points. Here differential equations in a variety of situations are considered: symmetry breaking, forcing, period doubling, twisted orbits, first integrals, gradients etc. Periodic solutions of Hamiltonian systems, in particular spring-pendulum systems, are studied as well as Hopf bifurcation for all these situations.

Chern character in twisted K theory: Equivariant and holomorphic cases

Abstract: It was argued in [25, 5] that in the presence of a nontrivial B-field, D-brane charges in type IIB string theories are classified by twisted K-theory. In [4], it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary Hilbert bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal projective unitary bundle. The principal projective unitary bundle is in turn determined by the twist. This paper studies in detail the Chern-Weil representative of the Chern character of bundle gerbe K-theory that was introduced in [4], extending the construction to the equivariant and the holomorphic cases. Included is a discussion of interesting examples.

Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions

Abstract: Holder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion). Using an equivariant version of the Ruelle transfer operator studied by Parry and Pollicott, we obtain similar results for equivariant observations on compact group extensions of hyperbolic basic sets.

Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds

Abstract: Let W be a compact manifold and let \rho be a representation of its fundamental group into PSL(2,C). The volume of \rho is defined by taking any \rho-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back of the hyperbolic volume form on a fundamental domain. It turns out that such a volume does not depend on the choice of the equivariant map. Dunfield extended this construction to the case of a non-compact (cusped) manifold M, but he did not prove the volume is well-defined in all cases. We prove here that the volume of a representation is always well-defined and depends only on the representation. We show that this volume can be easily computed by straightening any ideal triangulation of M. We show that the volume of a representation is bounded from above by the relative simplicial volume of M. Finally, we prove a rigidity theorem for representations of the fundamental group of a hyperbolic manifold. Namely, we prove that if M is hyperbolic and vol(\rho)=vol(M) then \rho is discrete and faithful.

Equivariant Weierstrass Preparation and Values of L-functions at Negative Integers

Abstract: We apply an equivariant version of the p-adic Weierstrass Preparation Theorem in the context of possible non-commutative gen- eralizations of the power series of Deligne and Ribet. We then con- sider CM abelian extensions of totally real fields and by combining our earlier considerations with the known validity of the Main Con- jecture of Iwasawa theory we prove, modulo the conjectural vanishing of certain µ-invariants, a (corrected version of a) conjecture of Snaith and the 'rank zero component' of Kato's Generalized Iwasawa Main Conjecture for Tate motives of strictly positive weight. We next use the validity of this case of Kato's conjecture to prove a conjecture of Chinburg, Kolster, Pappas and Snaith and also to compute ex- plicitly the Fitting ideals of certain naturalcohomology groups in terms of the values of Dirichlet L-functions at negative integers. This computation improves upon results of Cornacchia and Ostvaer, of Kurihara and of Snaith, and, modulo the validity of a certain aspect of the Quillen-Lichtenbaum Conjecture, also verifies a finer and more general version of a well known conjecture of Coates and Sinnott.

Intersection Cohomology on Nonrational Polytopes

Abstract: We consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image Rπ* (π is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf \(\mathcal{L}_\Phi\), called the minimal sheaf plays the role of an equivariant intersection cohomology complex on the corresponding toric variety (which exists if Φ is rational). Using \(\mathcal{L}_\Phi\) we define the intersection cohomology space IH(Φ). It is conjectured that a strictly convex piecewise linear function on Φ acts as a Lefschetz operator on IH(Φ). We show that this conjecture implies Stanley's conjecture on the unimodality of the generalized h-vector of a convex polytope.

Equivariant spectral triples

Abstract: We present the review of noncommutative symmetries applied to Connes’ formulation of spectral triples. We introduce the notion of equivariant spectral triples with Hopf algebras as isometries of noncommutative manifolds, relate it to other elements of theory (equivariant K-theory, homology, equivariant differential algebras) and provide several examples of spectral triples with their isometries: isospectral (twisted) deformations (including noncommutative torus) and finite spectral triples.

Equivariant Epsilon Constants, Discriminants and Étale Cohomology

Abstract: Let $L/K$ be a finite Galois extension of number fields. We formulate and study a conjectural equality between an element of the relative algebraic K-group $K_0(\mathbb{Z}[\mathrm{Gal}(L/K)], \mathbb{R})$ which is constructed from the equivariant Artin epsilon constant of $L/K$ and a sum of structural invariants associated to $L/K$. The precise conjecture is motivated by the requirement that a special case of the equivariant refinement of the Tamagawa Number Conjecture of Bloch and Kato (as formulated by Flach and the second-named author) should be compatible with the functional equation of the associated L-function. We show that, more concretely, our conjecture also suggests a completely systematic refinement of the central approach and results of classical Galois module theory. In particular, the evidence for our conjecture that we present here already strongly refines many of the main results of Galois module theory.