Friday, 21 February, 2014
SPEAKER: Prof. Chaim Goodman-Strauss, University of Arkansas
TITLE: Self-assembly of hierarchical non-periodic tilings
ABSTRACT: Aperiodic sets of tiles have been known for nearly fifty years -- such sets of tiles admit tilings of the plane, but remarkably, only non-periodic tilings, tilings in which, somehow, translational symmetry is disrupted at all scales. The most famous example, the Penrose tiles, bears a more than passing resemblance to many physical "quasicrystals", first discovered by Schechtman in 1982. However, local-assembly procedures, which presumably govern physical examples, proved elusive and this line of inquiry appeared closed with Dworkin and Shieh's 1995 result that any tiling with the "local isomorphism property" must include tilings with "deceptions" of arbitrary size. This result was widely interpreted as proving that local assembly procedures could never exist for any aperiodic set of tiles.
In fact this is far from the case. We give, for any tiling substitution system satisfying mild and general conditions, and any regular language of hierarchies of supertiles arising in this system, a set of tiles in the well-studied "Abstract Tile Assembly Model" such that every infinite configuration these tiles assemble into reduces to such a hierarchy and every hierarchy arises as such an infinite self-assembled configuration.
Refreshments will be available at 3:10 p.m.