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Aperiodicity, rotational tiling spaces and topological space groups (2021)
Journal Article
Hunton, J., & Walton, J. (2021). Aperiodicity, rotational tiling spaces and topological space groups. Advances in Mathematics, 388, Article 107855. https://doi.org/10.1016/j.aim.2021.107855

We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperiodic patterns have largely concentrated just on translational structure... Read More about Aperiodicity, rotational tiling spaces and topological space groups.

Chaotic Delone Sets (2021)
Journal Article
Alvarez Lopez, J. A., Barral Lijo, R., Hunton, J., Nozawa, H., & Parker, J. R. (2021). Chaotic Delone Sets. Discrete and Continuous Dynamical Systems - Series A, 41(8), 3781-3796. https://doi.org/10.3934/dcds.2021016

We present a definition of chaotic Delone set and establish the genericity of chaos in the space of (ϵ,δ)-Delone sets for ϵ≥δ. We also present a hyperbolic analogue of the cut-and-project method that naturally produces examples of chaotic Delone sets... Read More about Chaotic Delone Sets.

The homology core of matchbox manifolds and invariant measures (2018)
Journal Article
Clark, A., & Hunton, J. (2019). The homology core of matchbox manifolds and invariant measures. Transactions of the American Mathematical Society, 371(3), 1771-1793. https://doi.org/10.1090/tran/7398

We consider the topology and dynamics associated with a wide class of matchbox manifolds, including spaces of aperiodic tilings and suspensions of higher rank (potentially nonabelian) group actions on zero-dimensional spaces. For such a space we intr... Read More about The homology core of matchbox manifolds and invariant measures.

Topological Invariants for Tilings (2017)
Presentation / Conference Contribution
Hunton, J. (2017, October). Topological Invariants for Tilings. Presented at Spectral Structures and Topological Methods in Mathematical Quasicrystals, Oberwolfach, Germany

The mathematical theory of aperiodic order grew out of various predecessors in discrete geometry, harmonic analysis and mathematical physics, and developed rapidly after the discovery of real world quasicrystals in 1982 by Shechtman. Many mathematica... Read More about Topological Invariants for Tilings.

Spaces of Projection Method Patterns and their Cohomology (2015)
Book Chapter
Hunton, J. (2015). Spaces of Projection Method Patterns and their Cohomology. In J. Kellendonk, D. Lenz, & J. Savinien (Eds.), Mathematics of aperiodic order (105-135). Birkhäuser Verlag. https://doi.org/10.1007/978-3-0348-0903-0_4

We explain from the basics why the Čech cohomology of a tiling space can be realised in terms of group cohomology, and use this to explain how to compute the cohomology of a projection pattern.

Integral cohomology of rational projection method patterns (2013)
Journal Article
Hunton, J., Gähler, F., & Kellendonk, J. (2013). Integral cohomology of rational projection method patterns. Algebraic & geometric topology, 13(3), 1661-1708. https://doi.org/10.2140/agt.2013.13.1661

We study the cohomology and hence K–theory of the aperiodic tilings formed by the so called “cut and project” method, that is, patterns in d –dimensional Euclidean space which arise as sections of higher dimensional, periodic structures. They form on... Read More about Integral cohomology of rational projection method patterns.