Showing papers on "Equivariant map published in 2001"

Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex.

Abstract: This paper is concerned with a striking visual experience: that of seeing geometric visual hallucinations. Hallucinatory images were classified by Kluver into four groups called form constants comprising (i) gratings, lattices, fretworks, filigrees, honeycombs and chequer-boards, (ii) cobwebs, (iii) tunnels, funnels, alleys, cones and vessels, and (iv) spirals. This paper describes a mathematical investigation of their origin based on the assumption that the patterns of connection between retina and striate cortex (henceforth referred to as V1)-the retinocortical map-and of neuronal circuits in V1, both local and lateral, determine their geometry. In the first part of the paper we show that form constants, when viewed in V1 coordinates, essentially correspond to combinations of plane waves, the wavelengths of which are integral multiples of the width of a human Hubel-Wiesel hypercolumn, ca. 1.33-2 mm. We next introduce a mathematical description of the large-scale dynamics of V1 in terms of the continuum limit of a lattice of interconnected hypercolumns, each of which itself comprises a number of interconnected iso-orientation columns. We then show that the patterns of interconnection in V1 exhibit a very interesting symmetry, i.e. they are invariant under the action of the planar Euclidean group E(2)-the group of rigid motions in the plane-rotations, reflections and translations. What is novel is that the lateral connectivity of V1 is such that a new group action is needed to represent its properties: by virtue of its anisotropy it is invariant with respect to certain shifts and twists of the plane. It is this shift-twist invariance that generates new representations of E(2). Assuming that the strength of lateral connections is weak compared with that of local connections, we next calculate the eigenvalues and eigenfunctions of the cortical dynamics, using Rayleigh-Schrodinger perturbation theory. The result is that in the absence of lateral connections, the eigenfunctions are degenerate, comprising both even and odd combinations of sinusoids in straight phi, the cortical label for orientation preference, and plane waves in r, the cortical position coordinate. 'Switching-on' the lateral interactions breaks the degeneracy and either even or else odd eigenfunctions are selected. These results can be shown to follow directly from the Euclidean symmetry we have imposed. In the second part of the paper we study the nature of various even and odd combinations of eigenfunctions or planforms, the symmetries of which are such that they remain invariant under the particular action of E(2) we have imposed. These symmetries correspond to certain subgroups of E(2), the so-called axial subgroups. Axial subgroups are important in that the equivariant branching lemma indicates that when a symmetrical dynamical system becomes unstable, new solutions emerge which have symmetries corresponding to the axial subgroups of the underlying symmetry group. This is precisely the case studied in this paper. Thus we study the various planforms that emerge when our model V1 dynamics become unstable under the presumed action of hallucinogens or flickering lights. We show that the planforms correspond to the axial subgroups of E(2), under the shift-twist action. We then compute what such planforms would look like in the visual field, given an extension of the retinocortical map to include its action on local edges and contours. What is most interesting is that, given our interpretation of the correspondence between V1 planforms and perceived patterns, the set of planforms generates representatives of all the form constants. It is also noteworthy that the planforms derived from our continuum model naturally divide V1 into what are called linear regions, in which the pattern has a near constant orientation, reminiscent of the iso-orientation patches constructed via optical imaging. The boundaries of such regions form fractures whose points of intersection correspond to the well-known 'pinwheels'. To complete the study we then investigate the stability of the planforms, using methods of nonlinear stability analysis, including Liapunov-Schmidt reduction and Poincare-Lindstedt perturbation theory. We find a close correspondence between stable planforms and form constants. The results are sensitive to the detailed specification of the lateral connectivity and suggest an interesting possibility, that the cortical mechanisms by which geometric visual hallucinations are generated, if sited mainly in V1, are closely related to those involved in the processing of edges and contours.

Semisimple Frobenius structures at higher genus

Abstract: In the context of equivariant Gromov-Witten theory of tori actions with isolated fixed points, we compute genus g > 1 Gromov-Witten potentials and their generalizations with gravitational descendents. Both formulas, with and without descendents, are stated in a form applicable to any semisimple Frobenius structure and therefore can be considered as definitions in the axiomatic context of Frobenius manifolds. In (nonequivariant) Gromov-Witten theory, they become conjectures expressing higher genus GW-invariants in terms of genus 0 GW-invariants of symplectic manifolds with generically semisimple quantum cup-product.

Puzzles and (equivariant) cohomology of Grassmannians

Abstract: We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown (abstractly) to have positive coefficients in [Graham] math.AG/9908172. Our formula is the first to be manifestly positive in this sense. In particular this gives a new and self-contained proof of the ordinary puzzle formula, by an induction backwards from the "most equivariant" case. The proof of the formula is mostly combinatorial, but requires no prior combinatorics, and only a modicum of equivariant cohomology (which we include). This formula is closely related to the one in [Molev-Sagan] q-alg/9707028 for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [Graham]. We include a cohomological interpretation of this problem, and a puzzle formulation for it.

Quiver varieties and tensor products

Abstract: In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety &?tilde; in a quiver variety, and show the following results: (1) The homology group of &?tilde; is a representation of a symmetric Kac-Moody Lie algebra ?, isomorphic to the tensor product V(λ1)⊗...⊗V(λ N ) of integrable highest weight modules. (2) The set of irreducible components of &?tilde; has a structure of a crystal, isomorphic to that of the q-analogue of V(λ1)⊗...⊗V(λ N ). (3) The equivariant K-homology group of &?tilde; is isomorphic to the tensor product of universal standard modules of the quantum loop algebra U q (L?), when ? is of type ADE. We also give a purely combinatorial description of the crystal of (2). This result is new even when N=1.

Open string Gromov-Witten invariants: Calculations and a mirror 'theorem'

Abstract: We propose localization techniques for computing Gromov-WitteninvariantsofmapsfromRiemannsurfaceswith boundariesintoaCalabi-Yau, with the boundaries mapped to a Lagrangian submanifold. Thecomputations can be expressed in terms of Gromov-Witten invariantsof one-pointed maps. In genus zero, an equivariant version of the mir-ror theorem allows us to write down a hypergeometric series, whichtogether with a mirror map allows one to compute the invariants to allorders, similar to the closed string model or the physics approach viamirror symmetry. In the noncompact example where the Calabi-Yau isK P 2 ,our results agree with physics predictions at genus zero obtainedusing mirror symmetry for open strings. At higher genera, our resultssatisfy strong integrality checks conjectured from physics. 1 Introduction 1.1 The Physics Mirror symmetry is famous for being able to predict Gromov-Witten invari-ants of Calabi-Yau manifolds. The basic conjecture is that there is a dualitybetween string theories on mirror Calabi-Yau manifolds. As a consequence,the topological field theory defined from the A-twist of one Calabi-Yau man-ifold is equal to the topological B-twist of the mirror. Both twists can beperformed on Calabi-Yau target manifolds. From a practical point of view,in order to obtain enumerative predictions, one needs to know the theoryon the B-model (in this case, defined through classical period integrals) aswell as an identification of the parameter spaces for both theories – the“mirror map.” To extract integer-valued invariants, one needs an all-genus“multiple-cover” formula. The technology for finding mirror manifolds [3]1

Formation of singularities for equivariant (2+1)-dimensional wave maps into the 2-sphere

Abstract: In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from (2+1)-dimensional Minkowski spacetime into the 2-sphere. Our results provide strong evidence for the conjecture that large-energy initial data develop singularities in finite time and that singularity formation has the universal form of adiabatic shrinking of the degree-one harmonic map from 2 into S2.

Dimensional reduction, sl(2,c)-equivariant bundles and stable holomorphic chains

Abstract: In this paper we study gauge theory on SL(2,C)-equivariant bundles overX×P 1 ,w here X is a compact Kahler manifold, P 1 is the complex projective line, and the action of SL(2, C) is trivial on X and standard on P 1 . We first classify these bundles, showing that they are in correspondence with objects onX — that we call holomorphic chains — consisting of a finite number of holomorphic bundles Ei and morphismsEi →E i 1. We then prove a Hitchin-Kobayashi correspondence relating the existence of solutions to certain natural gauge-theoretic equations and an appropriate notion of stability for an equivariant bundle and the corresponding chain. A central tool in this paper is a dimensional reduction procedure which allow us to go fromX×P 1 toX.

On Affine Equivariant Multivariate Quantiles

Abstract: An extension of univariate quantiles in the multivariate set-up has been proposed and studied The proposed approach is affine equivariant, and it is based on an adaptive transformation retransformation procedure Behadur type linear representations of the proposed quantiles are established and consequently asymptotic distributions are also derived As applications of these multivariate quantiles, we develop some affine equivariant quantile contour plots which can be used to study the geometry of the data cloud as well as the underlying probability distribution and to detect outliers These quantiles can also be used to construct affine invariant versions of multivariate Q-Q plots which are useful in checking how well a given multivariate probability distribution fits the data and for comparing the distributions of two data sets We illustrate these applications with some simulated and real data sets We also indicate a way of extending the notion of univariate L-estimates and trimmed means in the multivariate set-up using these affine equivariant quantiles

Equivariant Seiberg-Witten Floer Homology

Abstract: In this paper we construct, for all compact oriented three- manifolds, a U(1)-equivariant version of Seiberg-Witten Floer homology, which is invariant under the choice of metric and perturbation. We give a detailed analysis of the boundary structure of the monopole moduli spaces, compactified to smooth manifolds with corners. The proof of the independence of metric and perturbation is then obtained via an analysis of all the relevant obstruction bundles and sections, and the corresponding gluing theorems. The paper also contains a discussion of the chamber structure for the Seiberg-Witten invariant for rational homology 3-spheres, and proofs of the wall crossing formula, obtained by studying the exact sequences relating the equivariant and the non-equivariant Floer homologies and by a local model at the reducible monopole.

Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic

Abstract: We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the "algebraic" deformation functor of an ordinary curve X over a field of positive characteristic with prescribed action of a finite group G, and the data are computed in terms of the ramification behaviour of X → G\X. The second one is the "analytic" deformation functor of a fixed embedding of a finitely generated discrete group N in P GL(2, K) over a non-archimedean valued field K, and the data are computed in terms of the Bass-Serre representation of N via a graph of groups. Finally, if is a free subgroup of N such that N is contained in the normalizer of in P GL(2, K), then the Mumford curve associated to becomes equipped with an action of N/, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of N.

Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K-theory

Abstract: We introduce the notion of generalized orbifold Euler charac- teristic associated to an arbitrary group, and study its properties. We then calculate generating functions of higher order (p-primary) orbifold Euler characteristic of symmetric products of a G-manifold M. As a corollary, we obtain a formula for the number of conjugacy classes of d-tuples of mu- tually commuting elements (of order powers of p) in the wreath product GoSn in terms of corresponding numbers of G. As a topological applica- tion, we present generating functions of Euler characteristic of equivariant Morava K-theories of symmetric products of a G-manifold M. AMS Classication 55N20, 55N91; 57S17, 57D15, 20E22, 37F20, 05A15

Conformally equivariant quantum Hamiltonians

Abstract: Let (M, g) be a pseudo-Riemannian manifold and $ \cal{F}_\lambda(M) $ the space of densities of degree $ \lambda $ on M. We study the space $ \cal{D}^2_{\lambda,\mu}(M) $ of second-order differential operators from $ \cal{F}_\lambda(M) $ to $ \cal{F}_\mu(M) $ . If (M, g) is conformally flat with signature p - q, then $ \cal{D}^2_{\lambda,\mu}(M) $ is viewed as a module over the group of conformal transformations of M. It turns out that, for almost all values of $ \mu-\lambda $ , the O(p+1, q+1)-modules $ \cal{D}^2_{\lambda,\mu}(M) $ and the space of symbols (i.e., of second-order polynomials on $ T^*M $ ) are canonically isomorphic. This yields a conformally equivariant quantization for quadratic Hamiltonians. We furthermore show that this quantization map extends to arbitrary pseudo-Riemannian manifolds and depends only on the conformal class [g] of the metric. As an example, the quantization of the geodesic flow yields a novel conformally equivariant Laplace operator on half-densities, as well as the well-known Yamabe Laplacian. We also recover in this framework the multi-dimensional Schwarzian derivative of conformal transformations.

DIMENSIONAL REDUCTION, ${\rm SL} (2, {\mathbb C})$-EQUIVARIANT BUNDLES AND STABLE HOLOMORPHIC CHAINS

Abstract: In this paper we study gauge theory on ${\rm SL} (2, {\mathbb C})$-equivariant bundles over X × ℙ1, where X is a compact Kahler manifold, ℙ1 is the complex projective line, and the action of ${\rm SL} (2, {\mathbb C})$ is trivial on X and standard on ℙ1. We first classify these bundles, showing that they are in correspondence with objects on X — that we call holomorphic chains — consisting of a finite number of holomorphic bundles ℰi and morphisms ℰi → ℰi-1. We then prove a Hitchin–Kobayashi correspondence relating the existence of solutions to certain natural gauge-theoretic equations and an appropriate notion of stability for an equivariant bundle and the corresponding chain. A central tool in this paper is a dimensional reduction procedure which allow us to go from X × ℙ1 to X.

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.

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).

A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof

Abstract: We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic K 0-theory for these varieties. We use the equivariant analytic torsion to define direct image maps in this context and we prove a Riemann-Roch theorem for the natural transformation of equivariant arithmetic K 0-theory induced by the restriction to the fixed point scheme; this theorem can be viewed as an analog, in the context of Arakelov geometry, of the regular case of the theorem proved by P. Baum, W. Fulton and G. Quart in [BaFQ]. We show that it implies an equivariant refinement of the arithmetic Riemann-Roch theorem, in a form conjectured by J.-M. Bismut (cf. [B2, Par. (l), p. 353] and also Ch. Soule’s question in [SABK, 1.5, p. 162]).

Chern characters for the equivariant K -theory of proper G - CW -complexes

Abstract: We first construct a classifying space for defining equivariant K-theory for proper actions of discrete groups. This is then applied to construct equivariant Chern characters with values in Bredon cohomology with coefficients in the representation ring functor R(—)(tensored by the rationals). And this in turn is applied to prove some versions of the Atiyah-Segal completion theorem for real and complex K-theory in this setting.

Finite volume ∞ows and Morse theory

Abstract: In this paper we present a new approach to Morse theory based on the de Rham-Federer theory of currents. The full classical theory is derived in a transparent way. The methods carry over uniformly to the equivariant and the holomorphic settings. Moreover, the methods are substantially stronger than the classical ones and have interesting applications to geometry. They lead, for example, to formulas relating characteristic forms and singularities of bundle maps.

Computations of complex equivariant bordism rings

Abstract: We give explicit computations of the coefficients of homotopical complex equivariant cobordism theory MU G , when G is abelian. We present a set of generators which is complete for any abelian group. We present a set of relations which is complete when G is cyclic and which we conjecture to be complete in general. We proceed by first computing the localization of MU G obtained by inverting Euler classes of representations. We then define a family of operations which essentially divide by Euler classes and use these operations to define our generating sets. We give geometric applications of these computations to the study of equivariant genera, circle actions on four-manifolds, and cobordism relations between Lens spaces.

Triples, Fluxes, and Strings

Abstract: We study string compactifications with sixteen supersymmetries. The moduli space for these compactifications becomes quite intricate in lower dimensions, partly because there are many different irreducible components. We focus primarily, but not exclusively, on compactifications to seven or more dimensions. These vacua can be realized in a number ways: the perturbative constructions we study include toroidal compactifications of the heterotic/type I strings, asymmetric orbifolds, and orientifolds. In addition, we describe less conventional M and F theory compactifications on smooth spaces. The last class of vacua considered are compactifications on singular spaces with non-trivial discrete fluxes. We find a number of new components in the string moduli space. Contained in some of these components are M theory compactifications with novel kinds of ``frozen'' singularities. We are naturally led to conjecture the existence of new dualities relating spaces with different singular geometries and fluxes. As our study of these vacua unfolds, we also learn about additional topics including: F theory on spaces without section, automorphisms of del Pezzo surfaces, and novel physics (and puzzles) from equivariant K-theory. Lastly, we comment on how the data we gain about the M theory three-form might be interpreted.

Equivariant elliptic cohomology and rigidity

Abstract: Equivariant elliptic cohomology with complex coefficients was defined axiomatically by Ginzburg, Kapranov and Vasserot and constructed by Grojnowski. We give an invariant definition of complex S 1 -equivariant elliptic cohomology, and use it to give an entirely cohomological proof of the rigidity theorem of Witten for the elliptic genus. We also state and prove a rigidity theorem for families of elliptic genera.

Projectively Equivariant Quantization and Symbol Calculus: Noncommutative Hypergeometric Functions

Abstract: We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on S3.

Towards Projectively Equivariant Quantization

Abstract: In this paper, we define a natural generalization of the notion of projectively equivariant quantization on a flat space to arbitrary manifolds equipped with arbitrary projective structures. We show how to get such a quantization for the differential operators of order 2 and explain how the method could be adapted to construct a quantization for all the differential operators. In particular, we state a conjecture about the relevant cohomologies that would insure the existence of such quantization in all cases.

On Grassmannians associated with JB^*-triples

Abstract: Then Γ is a complex Lie group that has precisely n+ 1 orbits E0, . . . , En in E – each Er is the (locally closed) complex submanifold of all matrices of rank r. Clearly, Er−1 is in the closure of Er for every 1 ≤ r ≤ n and En is the unique open orbit. For every pair of integers p, q ≥ 0 let Gp,q be the Grassmannian of all p-planes in Cp+q. Then Gp,q is a compact symmetric hermitian manifold of dimension pq holomorphically equivalent to Gq,p. Every Er is in a canonical way a Γ -equivariant holomorphic fibre bundle

Projective modules of finite type over the supersphere S2,2

Abstract: In the spirit of noncommutative geometry we construct all inequivalent superline bundles over the (2,2)-dimensional supersphere S2,2 by means of global projectors p via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding “rank 1” supervector bundle over S2,2. The canonical connection ∇=p∘d is used to compute the Chern numbers by means of the Berezin integral on S2,2. The associated connection 1-forms are graded extensions of monopoles with not trivial topological charge. Supertransposed projectors gives opposite values for the charges. We also comment on the K-theory of S2,2.

Calogero–Moser Systems and Hitchin Systems

Abstract: We exhibit the elliptic Calogero–Moser system as a Hitchin system of G-principal Higgs pairs. The group G, though naturally associated to any root system, is not semi-simple. We then interpret the Lax pairs with spectral parameter of d'Hoker and Phong [dP1] and Bordner, Corrigan and Sasaki [BCS1] in terms of equivariant embeddings of the Hitchin system of G into that of GL(N).

An algebraic model for rational S^1-equivariant stable homotopy theory

Abstract: Greenlees defined an abelian category A whose derived category is equivalent to the rational S^1-equivariant stable homotopy category whose objects represent rational S^1-equivariant cohomology theories. We show that in fact the model category of differential graded objects in A (dgA) models the whole rational S^1-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S^1-equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The new ingredients here are certain Massey product calculations and the work on rational stable model categories from "Classification of stable model categories" and "Equivalences of monoidal model categories" with Schwede. In an appendix we show that Toda brackets, and hence also Massey products, are determined by the triangulated derived category.