Applications of space filling curves to substitution tilings

One of the most spectacular scientific discoveries of the late twentieth century was a new material that was neither crystalline nor amorphous. This created a paradigm shift in crystallography, and these alloys are now called quasicrystals. Quasicrystals are modelled mathematically by patterns called aperiodic tilings that lack symmetry in the usual sense, but still exhibit long-range order. The most famous example is due to Sir Roger Penrose, whose tiling exhibited the same `impossible' symmetry as the quasicrystals discovered by Professor Dan Shechtman. The research in this proposal initiates a new method of studying aperiodic tilings through dimension reduction.

The digital revolution has made profound advances in sending two- and even three-dimensional images from place to place by encoding them as a sequence of zeros and ones. The research in this proposal draws analogy with this except that the image is infinite and need not have pixels arranged in a locally systematic way. In particular, the research in this proposal initiates a similar type of encoding of an aperiodic tiling through the use of space-filing curves; a type of fractal that was discovered in the late 18th century that helped to reshape our mathematical notions of size, area and volume. In much the same way as a video feed, some information is compressed through the encoding. However, we can still garner a vast amount of information about the original tiling, especially when the space filling curve comes from the underlying method used to define the tiling in the first place. Significantly, in the case of aperiodic tilings, all the geometric information about how tiles fit together is encoded in the digital sequence making it very easy to work with from a mathematical perspective; it is a purely combinatorial object.

To each aperiodic tiling we define a dynamical system that consists of a map on a topological space whose individual points are infinite tilings. It has been shown that this topological space is a Cantor set fibre bundle over a torus; that is, it is a donut with an arbitrary number of holes that has fractals emanating from every point on its surface. The bizarre nature of this space makes it extremely difficult to study. For this reason, topological and operator algebraic invariants have been the focus of research on tiling spaces. The programme of research outlined in this proposal gives a new attack on studying this dynamical system by studying the combinatorial space associated with the space filling curve, which is much simpler while retaining most information about the more complicated system.

The new approach taken in this proposal will have impact across research in aperiodic tiling theory, and even to the more general study of hyperbolic dynamical systems, operator algebras and fractal geometry.

A large part of the impact of this proposal comes from the cross-disciplinary nature of the research and results. The study of aperiodic tilings has connections with philosophy, mathematics and material sciences. Even within mathematics, the proposed research interfaces with topological dynamics, operator algebras and fractal geometry.

The academic impact of this action will be realised through the completely new method of studying aperiodic substitution tilings. In particular, researchers studying any aspect of aperiodic tilings will be able to use the outcomes of this proposal to investigate their particular area of research. On a broader scale, the research in this proposal will forge new directions in topological dynamics and operator algebras and strengthen the existing links between these two areas.

In the sciences, the proposed research could lead to a better understanding of certain metallic alloys called quasicrystals, and how they are formed. In computer science, space filling curves lead to efficient algorithms for writing data and the proposed research may impact this research program through the discovery of many new space filling curves and through their links with tilings.

Finally, the research in this proposal will help to maintain the international reputation of the United Kingdom as a leader in science. In particular, the funding in this proposal will be used to develop new talent through a postdoctoral fellowship in order to further advance the research leaders of tomorrow.