IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Fixed parameter extrapolation and quasicrystals in the plane

  • Oct. 14, 2013
  • 4 p.m.
  • LeConte 312

Abstract

Fix any complex number z. We say that a set S in the complex plane is z-convex if, whenever x and y are in S, the point (1-z)x + zy is also in S. We present a number of facts about z-convex sets and their (topological) closures. For example, letting Q_z be the least z-convex superset of {0,1}, it turns out that the closure of Q_z is either convex (in the usual sense) or else uniformly discrete. In the former case, the closure of Q_z is either the unit interval [0,1], the real line, or the whole complex plane, depending on z. On the other hand, we give many interesting examples of uniformly discrete Q_z, including aperiodic sets (with no translational symmetry). These are examples of Meyer sets, sometimes called quasicrystals. Some (but not all) previously known planar quasicrystals are z-convex, and we obtain types of aperiodic Meyer sets that (to the best of our knowledge) are new. We will show several pictures of these sets, especially ones generated from regular polygons.

This is joint work with Frederic Green (Clark University, Worcester, MA), Rohit Gurjar (IIT Kanpur, India), and Steven Homer (Boston University, Boston, MA).

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