On the Noncommutative Geometry of Tilings

Julien, Antoine; Kellendonk, Johannes; Savinien, Jean

doi:10.1007/978-3-0348-0903-0_8

Cited by 9 publications

(9 citation statements)

References 38 publications

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“…Problems of tiling (as in the case of protein folding) could be considered using a geometrical approach, i.e., without the use of physical concepts. The mathematical approaches used in tiling (Frank, 2015;Julien et al, 2015) may be useful for understanding the mechanisms of protein folding and for understanding the mechanisms of a wide class of biochemical reactions.…”

Section: Quasicrystal Protein In the Native Configurationmentioning

confidence: 99%

On a generalized Levinthal's paradox: The role of long- and short range interactions in complex bio-molecular reactions, including protein and DNA folding

Melkikh

Meijer

2018

Progress in Biophysics and Molecular Biology

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The current protein folding literature is reviewed. Two main approaches to the problem of folding were selected for this review: geometrical and biophysical. The geometrical approach allows the formulation of topological restrictions on folding, that are usually not taken into account in the construction of physical models. In particular, the topological constraints do not allow the known funnel-like energy landscape modeling, although most common methods of resolving the paradox are based on this method. The very paradox is based on the fact that complex molecules must reach their native conformations (complexes that result from reactions) in an exponentially long time, which clearly contradicts the observed experimental data. In this respect we considered the complexity of the reactions between ligands and proteins. On this general basis, the folding-reaction paradox was reformulated and generalized. We conclude that prospects for solving the paradox should be associated with incorporating a topology aspect in biophysical models of protein folding, through the construction of hybrid models. However, such models should explicitly include long-range force fields and local cell biological conditions, such as structured water complexes and photon/phonon/soliton waves, ordered in discrete frequency bands. In this framework, collective and coherent oscillations in, and between, macromolecules are instrumental in inducing intra- and intercellular resonance, serving as an integral guiding network of life communication: the electrome aspect of the cell. Yet, to identify the actual mechanisms underlying the bonds between molecules (atoms), it will be necessary to perform dedicated experiments to more definitely solve the particular time paradox.

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Section: Quasicrystal Protein In the Native Configurationmentioning

confidence: 99%

On a generalized Levinthal's paradox: The role of long- and short range interactions in complex bio-molecular reactions, including protein and DNA folding

Melkikh

Meijer

2018

Progress in Biophysics and Molecular Biology

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“…The statement for (1+X 2 ) −δ [T, χ * k ] follows in a similar manner. As previously mentioned, it would be interesting to compare the K-homology class of the ε-unbounded Fredholm module from Theorem 5.14 with similar constructions in the tiling literature [64,57].…”

Section: Denote Bymentioning

confidence: 99%

“…Spectral triple constructions for C * r (G) building from the Pearson-Bellissard framework have already appeared in the tiling literature [57,64]. While the setting of each construction is quite different, it would be interesting to better understand the relationship between these spectral triples and our unbounded Fredholm module.…”

Section: Unbounded Fredholm Modules For Lattices With Finite Local Co...mentioning

confidence: 99%

Index Theory and Topological Phases of Aperiodic Lattices

Bourne

Mesland

2019

Ann. Henri Poincaré

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We examine the noncommutative index theory associated to the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials, and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.

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“…Since Bellissard and Pearson's seminal result there have been a number of papers on spectral triples of tilings, see for example [13,14,17,19]. The survey article [11] explains these constructions and their relationship to one another.…”

Section: Introductionmentioning

confidence: 99%

Fractal spectral triples on Kellendonk’sC∗-algebra of a substitution tiling

Mampusti

Whittaker

2017

Journal of Geometry and Physics

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Abstract. We introduce a new class of noncommutative spectral triples on Kellendonk's C * -algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using PerronFrobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is θ-summable, and respects the hierarchy of the substitution system. To elucidate our results, we construct a fractal tree on the Penrose tiling, and explicitly show how it gives rise to a collection of spectral triples.

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On the Noncommutative Geometry of Tilings

Cited by 9 publications

References 38 publications

On a generalized Levinthal's paradox: The role of long- and short range interactions in complex bio-molecular reactions, including protein and DNA folding

On a generalized Levinthal's paradox: The role of long- and short range interactions in complex bio-molecular reactions, including protein and DNA folding

Index Theory and Topological Phases of Aperiodic Lattices

Fractal spectral triples on Kellendonk’sC∗-algebra of a substitution tiling

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