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.
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.
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).
Queen reflections - a modification of Wythoff Nim (2022)
Rust, Dan and Fokkink, Robbert
International Journal of Game Theory ((In Press))
Measure theoretic entropy of random substitution subshifts (2022)
Gohlke, Philipp; Mitchell, Andrew; Rust, Dan and Samuel, Tony
Annales Henri Poincaré ((Early Access))
Topological mixing of random substitutions (2021)
Miro, Eden; Rust, Dan; Sadun, Lorenzo and Tadeo, Gwendolyn S.
Periodic points in random substitution subshifts (2020-11)
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 (2016-06-01)
Topology and its Applications, 205 (pp. 58-81)