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We introduce the use of "regular production systems"--- a certain generalization of symbolic substitution systems--- as a tool for analyzing tilings in general. These systems precisely capture the combinatorial structure of any set of tiles residing on a two-dimensional surface, though in this talk we are particularly interested in tilings of the hyperbolic plane. We briefly discuss a number of applications, such as the construction of the first known "strongly aperiodic" set of tiles in the hyperbolic plane. As another application, we conjecture necessary and sufficient conditions under which we may tile the sphere, hyperbolic or Euclidean plane by copies of a given triangle, and prove the conjecture on all but a measure-zero set in the space of all triangles. We give a new proof of Poincare's Triangle theorem as an aside. We also show most triangles that do tile are "weakly aperiodic"; that is, they admit tilings, and admit tilings that are invariant under some infinite cyclic symmetry, but do not admit tilings with a compact fundamental domain. Decidability and rigidity play an interesting role.

Join seminar with the Geometry Topology seminar.