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Dr Dan Rust

Profile summary

Professional biography

I am a Lecturer of Applied Mathematics at the Open University. After receiving an MMath from Manchester University in 2012, I obtained my PhD from the University of Leicester under the supervision of Alex Clark. I then undertook a 4 year position in Bielefeld Unviersity as a postdoctoral researcher as a member of the Aperiodic Order group of Michael Baake. I joined the Open University from there in May 2020.

Research interests

My research is focused on symbolic dynamics, as well as the topology, dynamics and spectral theory of aperiodic tilings and quasicrystals. Specific topics include:

  • The representation of tiling spaces as inverse limits of branched manifolds and their Čech cohomology

  • The dynamics of tilings and bi-infinite sequences - especially those arising from substitutions, S-adic systems and random substitutions, as well as cut and project sets

  • Applications of aperiodic order to other areas of mathematics - specifically the study of combinatorial game theory, dimension groups and Diophantine approximation

I also have interests in general topological dynamics and algebraic topology as a whole. I have a particular fondness for the applications of algebraic topology to knot theory and braids.

The above image is a Barge Diamond complex which was automatically generated by the program Grout.


Developed by myself and Scott Balchin, Grout is a free user-friendly program for Windows and Mac which can calculate various topological and combinatorial properties of symbolic substitutions.
For an explanation of the computations that Grout performs, we have provided documentation on the arXiv.
You can download Grout from Github.

Teaching interests

I am involved in the production and presentation of several undergraduate and postgraduate modules in Mathematics, including MST124 - Essential Mathematics 1, MST326 - Mathematical Methods and Fluid Mechanics, MST374 - Computational Applied Mathematics, M823 - Analytic Number Theory I and M840 - Dissertation in Mathematics (Topic: Aperiodic Tilings and Symbolic Dynamics).


A class of aperiodic honeycombs with tuneable mechanical properties (2024-04)
Moat, Richard J; Clarke, Daniel John; Carter, Francesca; Rust, Dan and Jowers, Iestyn
Applied Materials Today, 37, Article 102127

Spectral properties of substitutions on compact alphabets (2023-10)
Mañibo, Neil; Rust, Dan and Walton, James J.
Bulletin of the London Mathematical Society, 55(5) (pp. 2425-2445)

Measure theoretic entropy of random substitution subshifts (2023-01)
Gohlke, P.; Mitchell, A.; Rust, D. and Samuel, T.
Annales Henri Poincaré, 24(1) (pp. 277-323)

Automorphism groups of random substitution subshifts (2023)
Fokkink, Robbert; Rust, Dan and Salo, Ville
Indagationes Mathematicae ((early access))

Topological mixing of random substitutions (2022-11-28)
Miro, Eden; Rust, Dan; Sadun, Lorenzo and Tadeo, Gwendolyn S.
Israel Journal of Mathematics(2022)

Queen reflections: a modification of Wythoff Nim (2022)
Fokkink, Robbert and Rust, Dan
International Journal of Game Theory ((Early Access))

Periodic points in random substitution subshifts (2020-11)
Rust, Dan
Monatshefte für Mathematik, 193 (pp. 683-704)

Shifts of finite type and random substitutions (2019-09)
Gohlke, Philipp; Rust, Dan and Spindeler, Timo
Discrete & Continuous Dynamical Systems - A, 39(9) (pp. 5085-5103)

Dynamical systems arising from random substitutions (2018-08)
Rust, Dan and Spindeler, Timo
Indagationes Mathematicae, 29(4) (pp. 1131-1155)

Beyond primitivity for one-dimensional substitution subshifts and tiling spaces (2018-05)
Maloney, Gregory R. and Rust, Dan
Ergodic Theory and Dynamical Systems, 38(3) (pp. 1086-1117)

Computations for Symbolic Substitutions (2017-01-27)
Balchin, Scott and Rust, Dan
Journal of Integer Sequences, 20, Article 17.4.1

An uncountable set of tiling spaces with distinct cohomology (2016)
Rust, Dan
Topology and its Applications, 205 (pp. 58-81)