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A typical crystal is a finite piece of a material which may be invariant under some point symmetry group. If it is a so-called intrinsic higher-order topological insulator or superconductor, then it displays boundary modes at hinges or corners protected by the crystalline symmetry and the bulk topology. We explain the mechanism behind such phenomena using operator K-theory. Specifically, we derive a groupoid C ∗ -algebra that (1) encodes the dynamics of the electrons in the infinite size limit of a crystal; (2) remembers the boundary conditions at the crystal’s boundaries, and (3) admits a natural action by the point symmetries of the atomic lattice. The filtrations of the groupoid’s unit space by closed subsets that are invariant under the groupoid and point group actions supply equivariant cofiltrations of the groupoid C ∗ -algebra. We show that specific derivations of the induced spectral sequences in twisted equivariant K-theories enumerate all non-trivial higher-order bulk-boundary correspondences.
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This content will become publicly available on March 3, 2026
Two results on the Convex Algebraic Geometry of sets with continuous symmetries
Abstract We prove two results on convex subsets of Euclidean spaces invariant under an orthogonal group action. First, we show that invariant spectrahedra admit an equivariant spectrahedral description, that is, can be described by an equivariant linear matrix inequality. Second, we show that the bijection induced by Kostant's Convexity Theorem between convex subsets invariant under a polar representation and convex subsets of a section invariant under the Weyl group preserves the classes of convex semialgebraic sets, spectrahedral shadows, and rigidly convex sets.
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- Award ID(s):
- 2142575
- PAR ID:
- 10576500
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 57
- Issue:
- 5
- ISSN:
- 0024-6093
- Format(s):
- Medium: X Size: p. 1388-1408
- Size(s):
- p. 1388-1408
- Sponsoring Org:
- National Science Foundation
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