It has just been announced that this year’s Nobel Prize for Chemistry goes to Daniel Schechtman, at Technion, for the discovery of the structure of quasicrystals.
A nice short overview of the topic and of the prize winner achievements can be found on today’s Nature News article.
Besides their importance in chemistry, quasicrystal structures have attracted a lot of attention from mathematicians and mathematical physicists, because of the particular property of the spectra of Schrödinger operators on such quasi-periodic structures.
Geometrically, quasi-crystals behave very much like Penrose tilings and, as such, they fit well within the kind of objects that can be treated by noncommutative geometry methods.
There is a substantial literature on quasicrystal and noncommutative geometry, so I am just going to list here a couple of my favorite papers on the topic, for those who may be interested in looking at what has been done with this geometric viewpoint.
– J.Bellissard, B.Iochum, E.Scoppola, D.Testard, “Spectral properties of one-dimensional quasi-crystals”. Comm. Math.Phys. 125 (1989) N.3, 527-543.
– J.Bellissard, D.J.L. Herrmann, M. Zarrouati, “Hulls of aperiodic solids and gap labeling theorems”. Directions in mathematical quasicrystals, 207–258, CRM Monogr. 13, Amer. Math. Soc., Providence, RI, 2000.
– J. Bellissard, “The noncommutative geometry of aperiodic solids”. Geometric and topological methods for quantum field theory (Villa de Leyva, 2001), 86–156, World Sci. Publ., River Edge, NJ, 2003
– M.T. Benameur, H. Oyono-Oyono, “Index theory for quasi-crystals. I. Computation of the gap-label group”, J. Funct. Anal. 252 (2007) N.1, 137-170
Also a book I especially like on quasicrystals (though from a more physical and less mathematical perspective) is this: