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Equivariant map
About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.
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TL;DR: The Atiyah-Hirzebruch spectral sequence as discussed by the authors relates ordinary singular cohomology to complex K-theory, in a way that is explained further in this paper.
Abstract: In recent years much attention has been given to a certain spectral sequence relating motivic cohomology to algebraic K-theory [Be, BL, FS, V3]. This spectral sequence takes on the form H(X,Z(− q 2 )) ⇒ K(X), where the H(X ;Z(t)) are the bi-graded motivic cohomology groups, and K(X) denotes the algebraic K-theory of X . It is useful in our context to use topologists’ notation and write K(X) for what K-theorists call K−n(X). The above spectral sequence is the analog of the classical Atiyah-Hirzebruch spectral sequence relating ordinary singular cohomology to complex K-theory, in a way that is explained further below. It is well known that there are close similarities between motivic homotopy theory and the equivariant homotopy theory of Z/2-spaces (cf. [HK1, HK2], for example). In fact there is even a forgetful map of the form (motivic homotopy theory over R) → (Z/2-equivariant homotopy theory),
101 citations
TL;DR: The simplicial depth D(x), which is the probability that x be inside a random simplex whose vertices are p + 1 independent observations from F, is introduced and gives rise to a natural ordering of the data points from the center outward.
Abstract: For a distribution F on Rp and a point x in Rp the simplicial depth D(x), which is the probability that x be inside a random simplex whose vertices are p + 1 independent observations from F, is introduced. D(x) can be viewed as a measure of depth of the point x relative to F, and its empirical version gives rise to a natural ordering of the data points from the center outward. This ordering provides an approach for detecting outliers in a multivariate data cloud and leads to the introduction of affine equivariant multivariate generalizations of the univariate sample median and L-statistics. This sample median is shown to be consistent for the center of any angularly symmetric distribution.
101 citations
100 citations
TL;DR: In this paper, the torus-equivariant cohomology ring of isotropic Grassmannians is described by using a localization map to the Torus fixed points, and two types of formulas for equivariant Schubert classes of these homogeneous spaces are presented.
Abstract: We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call "excited Young diagrams", and the second one is written in terms of factorial Schur Q- or P-functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety.
100 citations
TL;DR: In this paper, the authors formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or non-commutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly non commutative) manifold satisfying certain regularity assumptions.
Abstract: We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. The idea of ‘quantum families’ (due to Woronowicz and Soltan) are relevant to our construction. A number of explicit examples are given and possible applications of our results to the problem of constructing quantum group equivariant spectral triples are discussed.
100 citations