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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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30 May 2014
TL;DR: In this article, Ricci flow with surgery and surger limits are discussed. But they do not cover the Gromov-Hausdorff convergence of Alexandrov spaces 2-dimensional Alexandrov Spaces 3-dimensional analogues.
Abstract: Introduction Geometric and analytic results for Ricci flow with surgery Ricci flow with surger Limits as t?? Local results valid for large time Proofs of the three propositions Locally volume collapsed 3-manifolds Introduction to part II The collapsing theorem Overview of the rest of the argument Basics of Gromov-Hausdorff convergence Basics of Alexandrov spaces 2-dimensional Alexandrov spaces 3-dimensional analogues The global result The equivariant case The equivariant case Bibliography Glossary of symbols Index

63 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the cyclic lamplighter groups C 2 ≀ C n embed into L 1 with uniformly bounded distortion, answering a question posed by Lee, Naor, and Peres in (26).
Abstract: Let G be a group generated by a finite set S and equipped with the associated left-invariant word metric d G . For a Banach space X, let α * X (G) (respectively, α / # X (G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively, an equivariant mapping) f:G → X and c > 0 such that for all x, y ∈ G we have ‖ f(x) − f(y)‖ ≥ c · d G (x, y) α . In particular, the Hilbert compression exponent (respectively, the equivariant Hilbert compression exponent) of G is (respectively, ). We show that if X has modulus of smoothness of power type p, then . Here β * (G) is the largest β ≥ 0 for which there exists a set of generators S of G and c > 0, such that for all we have , where { W t } ∞ t=0 is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X = L p , generalizes a theorem of Guentner and Kaminker (20), and answers a question posed by Tessera (37). We also show that, if then . This improves the previous bound due to Stalder and Valette (36). We deduce that if we write and then and use this result to answer a question posed by Tessera in (37) on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C 2 ≀ C n embed into L 1 with uniformly bounded distortion, answering a question posed by Lee, Naor, and Peres in (26). Finally, we use these results to show that edge Markov type need not imply Enflo type.

63 citations

Journal ArticleDOI
TL;DR: In this article, the authors apply homological algebra techniques from non-commutative topology to bivariant K-theory and show how to approximate a category by an Abelian category in a canonical way, such that the homolog- ical concepts of the category reduce to the corresponding concepts in the category.
Abstract: Bivariant (equivariant) K-theory is the standard setting for non- commutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from homological algebra: phantom maps, exact chain complexes, projective resolutions, and derived functors. We introduce these notions and apply them to examples from bivariant K-theory. An important observation of Beligiannis is that we can approximate our category by an Abelian category in a canonical way, such that our homolog- ical concepts reduce to the corresponding ones in this Abelian category. We compute this Abelian approximation in several interesting examples, where it turns out to be very concrete and tractable. The derived functors comprise the second page of a spectral sequence that, in favourable cases, converges towards Kasparov groups and other interesting objects. This mechanism is the common basis for many different spectral sequences. Here we only discuss the very simple 1-dimensional case, where the spectral sequences reduce to short exact sequences.

63 citations

Journal ArticleDOI
TL;DR: In this article, K-theory invariants of pseudodifferential operators on manifolds with corners were computed and an equivariant index theorem for operators invariant with respect to an action was proved.
Abstract: We compute K-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of $ \Bbb {R}^k $ . We briefly discuss the relation between our results and the $ \eta $ -invariant.

63 citations

Journal ArticleDOI
TL;DR: FeFeigin, M.Jimbo, T.Miwa and E.Mukhin this article studied the spaces of based quasi-maps from the projective line P^1 to the flag variety of g and showed that these spaces are normal and in the case when g is simply laced they are also Gorenstein and have rational singularities.
Abstract: Let g be a semi-simple Lie algebra. In this paper we study the spaces of based quasi-maps from the projective line P^1 to the flag variety of g (it is well-known that their singularities are supposed to model the singularities of the so called semi-infinite Schubert varieties which are hard to define directly). In the first part of the paper we show that the above spaces are normal and in the case when g is simply laced they are also Gorenstein and have rational singularities. In the second part of the paper we compute the character of the ring of functions on the above spaces; in view of the above results this computation can be thought of as a computation of the (equivariant) K-theoretic J-function of the flag variety of g. We show that when g is simply laced the above characters satisfy the "fermionic recursion" version of the difference quantum Toda lattice (due to B.Feigin, E.Feigin, M.Jimbo, T.Miwa and E.Mukhin). As a byproduct we show that the equivariant K-theoretic J-function of the flag variety of a simply laced Lie algebra g is the universal eigen-function of the difference quantum Toda lattice, thus proving a conjecture of Givental and Lee. Some modification of this result is also shown to hold for non-simply laced g. We also discuss an extension of the above results to the case when g is an affine Lie algebra (this extension is conjectural except when g=sl(N)).

63 citations


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