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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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03 Mar 2014TL;DR: In this paper, the Schrodinger Map equation in 2 + 1 dimensions with values into S 2 was considered, and it was shown that Q is unstable in the energy space H 1.
Abstract: We consider the Schrodinger Map equation in 2 + 1 dimensions, with values into S2. This admits a lowest energy steady state Q, namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. We prove that Q is unstable in the energy space H1. However, in the process of proving this we also show that within the equivariant class Q is stable in a stronger topology X ⊂ H 1. © 2013 by the American Mathematical Society. All rights reserved.
34 citations
TL;DR: In this paper, the authors presented two continuous symmetry reduction methods for reducing high-dimensional dissipative flows to local return maps by slicing the state space locally in such a way that each group orbit of symmetry-equivalent points is represented by a single point.
34 citations
01 Jan 1982
TL;DR: In this article, the problem of equivariant algebraic topological K-theory of rings and its calculation for finite fields was discussed, with the main focus on the complex Adams conjecture, and the real analog was assumed without proof by tom Dieck and Graeme Segal.
Abstract: There are many ways that group actions enter into algebraic K-theory and there are various theories that fit under the rubric of our title. To anyone familiar with both equivariant topological K-theory and Quillen's original definition and calculations of algebraic K-theory, there is a perfectly obvious program for the definition of the equivariant algebraic K-theory of rings and its calculation for finite fields. While this program surely must have occurred to others, there are no published accounts and the technical details have not been worked out before. That part of the program which pertains to the complex Adams conjecture was outlined in a letter to one of us from Graeme Segal, and the real analog was assumed without proof by tom Dieck [10,11.3.8]. Negatively indexed equivariant K-groups were introduced by Loday [19]. From a topological point of view, one way of thinking about Quillen's original definition runs as follows. Let ~ be a topological group, perhaps discrete. One has a notion of a principal K-bundle and a classifying space B~ for such bundles. When H is discrete, a principal ~-bundle is just a covering (possibly with disconnected total space) with fibre and group ~. Given any increasing sequence of groups N with union H, we obtain an increasing sequence of classifying spaces n BH with union B~. We then think of B~ as a classifying space for stable n bundles. When the ~ are discrete, B~ may have desirable homology groups but n will have trivial higher homotopy groups. In the cases of interest, we can use the plus construction to convert B~ to a Hopf space (= H-space) (BN) + with the same homology. When ~n = GL(n,A), we define ~q(B~) + = Kq(A) for q > 0. There is a more structured way of looking at (B~) +. In practice, we have sum maps {9 :~ x ~ ÷ ~m+n and a corresponding Whitney sum of bundles. While this can m n he used to give B~ a product, it is generally not a Hopf space, although it is so
34 citations
TL;DR: In this article, the authors consider periodic and quasi-periodic solutions of the three-body problem with homogeneous potential from the point of view of equivariant calculus of variations.
Abstract: We consider periodic and quasi-periodic solutions of the three-body problem with homogeneous potential from the point of view of equivariant calculus of variations First, we show that symmetry groups of the Lagrangian action functional can be reduced to groups in a finite explicitly given list, after a suitable change of coordinates Then, we show that local symmetric minimizers are always collisionless, without any assumption on the group other than the fact that collisions are not forced by the group itself Moreover, we describe some properties of the resulting symmetric collisionless minimizers (Lagrange, Euler, Hill-type orbits and Chenciner–Montgomery figure-eights)
34 citations
TL;DR: In this paper, an equivariant version of Farber's topological complexity for spaces with a given compact group action is defined and studied, and several examples and computations serve to highlight the similarities and differences with the nonequivariant case.
Abstract: We define and study an equivariant version of Farber’s topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The relationship of these invariants with the equivariant Lusternik‐Schnirelmann category is given. Several examples and computations serve to highlight the similarities and differences with the nonequivariant case. We also indicate how the equivariant topological complexity can be used to give estimates of the nonequivariant topological complexity. 55M99, 57S10; 55M30, 55R91
34 citations