Equivariant map

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

The catenoid estimate and its geometric applications

Abstract: We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such manifolds can be isotoped to be minimal and iii) the “doublings” of the Clifford torus by Kapouleas–Yang can be constructed variationally by an equivariant min-max procedure. In higher dimensions we also prove that the width of manifolds with positive Ricci curvature is achieved by an index $1$ orientable minimal hypersurface.

The elliptic Weyl character formula

Abstract: We calculate equivariant elliptic cohomology of the partial flag variety G/H, where H ⊆ G are compact connected Lie groups of equal rank. We identify the RO(G)-graded coefficientsEll ∗ G as powers of Looijenga’s line bundle and prove that transfer along the map �: G/H −→ pt is calculated by the Weyl-Kac character formula. Treating ordinary cohomology, K-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [GR].

Recursive evaluation and iterative contraction of N-body equivariant features.

Abstract: Mapping an atomistic configuration to a symmetrized N-point correlation of a field associated with the atomic positions (e.g., an atomic density) has emerged as an elegant and effective solution to represent structures as the input of machine-learning algorithms. While it has become clear that low-order density correlations do not provide a complete representation of an atomic environment, the exponential increase in the number of possible N-body invariants makes it difficult to design a concise and effective representation. We discuss how to exploit recursion relations between equivariant features of different order (generalizations of N-body invariants that provide a complete representation of the symmetries of improper rotations) to compute high-order terms efficiently. In combination with the automatic selection of the most expressive combination of features at each order, this approach provides a conceptual and practical framework to generate systematically improvable, symmetry adapted representations for atomistic machine learning.

Exploiting symmetry in boundary element methods

Abstract: Linear operator equations $\mathcal {L}f = g$ are considered in the context of boundary element methods, where the operator $\mathcal {L}$ is equivariant, i.e., commutes with the actions of a given finite symmetry group. By introducing a generalization of Reynolds projectors, a decomposition of the identity operator is constructed, which in turn allows the decomposition of the problem $\mathcal {L}f = g$ into a finite number of symmetric subproblems. The data function g does not need to possess any symmetry properties. It is shown that analogous reductions are possible for discretizations. An explicit construction of the corresponding reduced system matrices is given. This effects a considerable reduction in the computational complexity. For example, in the case of the isometry group of the 3-cube, the computational complexity of a direct linear equation solver for full matrices is reduced by 99.65 percent. Specific decompositions of the identity are given for most of the significant finite isometry group...