nLab noncommutative topology of quasiperiodicity

Idea

Quasi-periodic geometries such as Penrose tilings and quasicrystals tend to admit useful descriptions by noncommutative topology/noncommutative geometry.

References

For Penrose tilings

Noncommutative topology of Penrose tilings:

• Alain Connes, Section 2.3 of: Noncommutative Geometry, Academic Press, San Diego, CA, 1994 (ISBN:9780080571751, pdf)

• Hervé Oyono-Oyonoa, Samuel Petite, $C^\ast$-algebras of Penrose hyperbolic tilings, Journal of Geometry and Physics Volume 61, Issue 2, February 2011, Pages 400-424 (doi:10.1016/j.geomphys.2010.09.019)

For quasicrystals

Discussion of the noncommutative topology/KK-theory of the Brillouin zone of quasi-crystals in the spirit of the K-theory classification of topological phases of matter:

• Jean Bellissard, The Noncommutative Geometry of Aperiodic Solids, in: Geometric and Topological Methods for Quantum Field Theory, pp. 86-156 (2003) (pdf, doi:10.1142/9789812705068_0002)

• Fonger Ypma, Quasicrystals, $C^\ast$-algebras and K-theory, 2005 (pdf)

• Ian F. Putnam, Non-commutative methods for the K-theory of $C^\ast$-algebras of aperiodic patterns from cut-and-project systems, Commun. Math. Phys. 294, 703–729 (2010) (pdf, doi:10.1007/s00220-009-0968-0)