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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: In this article, the Tian-Zelditch expansion on the circle bundle associated to a positive line bundle on a complex projective manifold is considered. But the authors focus on the case of compact tori.
Abstract: Let X be the circle bundle associated to a positive line bundle on a complex projective (or, more generally, compact symplectic) manifold. The Tian-Zelditch expansion on X may be seen as a local manifestation of the decomposition of the (generalized) Hardy space H(X) into isotypes for the S 1-action. More generally, given a compatible action of a compact Lie group, and under general assumptions guaranteeing finite dimensionality of isotypes, we may look for asymptotic expansions locally reflecting the equivariant decomposition of H(X) over the irreducible representations of the group. We focus here on the case of compact tori.

38 citations

Journal ArticleDOI
TL;DR: 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.
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.

38 citations

Journal ArticleDOI
TL;DR: It is proved the "folk theorem" that in any Γ-equivariant dynamical system on Rk the only flow-invariant subspaces are the fixed-point spaces of subgroups of Γ.
Abstract: Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetry-breaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Γ of permutations of the nodes ("cells"). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals — ones that are not fixed-point spaces — can occur for such networks. We also prove the "folk theorem" that in any Γ-equivariant dynamical system on Rk the only flow-invariant subspaces are the fixed-point spaces of subgroups of Γ.

38 citations

Posted Content
Brooke Shipley1
TL;DR: In this article, it was shown that the model category of differential graded objects in A (dgA) models the whole 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.

37 citations

Journal ArticleDOI
TL;DR: In this article, the authors discuss the relation of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds.

37 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
2023463
2022888
2021630
2020658
2019526