Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

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

Permutation Equivariant Models for Compositional Generalization in Language

Abstract: Humans understand novel sentences by composing meanings and roles of core language components. In contrast, neural network models for natural language modeling fail when such compositional generalization is required. The main contribution of this paper is to hypothesize that language compositionality is a form of group-equivariance. Based on this hypothesis, we propose a set of tools for constructing equivariant sequence-to-sequence models. Throughout a variety of experiments on the SCAN tasks, we analyze the behavior of existing models under the lens of equivariance, and demonstrate that our equivariant architecture is able to achieve the type compositional generalization required in human language understanding.

Applications of equivariant cohomology

Abstract: We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic quotients and to indices of transversally elliptic operators. In particular, we state a conjecture for the index of a transversally elliptic operator linked to a Hamiltonian action. In the last part, we describe algorithms for numerical computations of values of multivariate spline functions and of vector-partition functions of classical root systems.

GL(n) equivariant Minkowski valuations

Abstract: A classification of all continuous GL(n) equivariant Minkowski valuations on convex bodies in $\mathbb{R}^n$ is established. Together with recent results of F.E. Schuster and the author, this article therefore completes the description of all continuous GL(n) intertwining Minkowski valuations.