Wednesday, October 5, 2011

quasicrystals on their way to Stockholm

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:

Enjoy your aperiodic pastimes...