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- Dr Reem Yassawi

I am a lecturer in the School of Mathematics and Statistics, based in Milton Keynes.

I joined the Open University in 2019, and previously held positions at Trent University, Canada, and Université Claude-Bernard Lyon 1, France.

My research interests lie in symbolic dynamics, automata theory, number theory and their interactions. In particular I like thinking about invariants of substitution dynamical systems, Bratteli-Vershik and S-adic systems, invariant measures for cellular automata, and automatic sequences.

Role | Start date | End date | Funding source |
---|---|---|---|

Lead | 01 Jul 2021 | 30 Jun 2024 | EPSRC Engineering and Physical Sciences Research Council |

Euclidean symmetries are all around us in the natural world. Some of these symmetries are visible to the naked eye, such as the bilateral symmetry of a butterfly's wings. Other symmetries can be viewed via an electron microscope, such as the translation symmetries of a crystal. More subtle to describe are the symmetries of quasicrystals, the existence of which was doubted for much of the last century. Quasicrystals are crystalline structures which do not have the translational symmetry of a normal crystal. Quasicrystals have a hierarchical structure: patterns and structures which appear on small scales are reproduced on larger and larger scales. The first mathematical model of a quasicrystal was discovered by Sir Roger Penrose half a century ago. The Penrose tiling has reflectional symmetry, but it lacks a translational symmetry. A translationally symmetric tiling of two dimensional space must have either three-, four- or six-fold rotational symmetry. But the Penrose tiling has local five-fold rotational symmetry. Penrose's tiling is simply a mathematical model, which is not necessarily guaranteed to exist in the natural world. But in 1982, Daniel Schechtman discovered that pentagonal symmetry actually appears in nature, while studying a rapidly chilled molten mixture of aluminium and manganese under an electron microscope. For his work, he received the Nobel prize in 2011. Since the discovery of the Penrose tilings, mathematicians have discovered many ways to create such arrangements: There are infinitely many mathematical tilings of the plane which do not have translational symmetry. Confined to the kinds of building-blocks provided by nature, it is harder for scientists to create, or discover, these tilings. Two questions arise, which are complementary to one another. The first is, when are two mathematical tilings somehow equivalent, and the second is, which of these mathematical tilings can be realised in the world around us? Answering the first question can guide scientists investigating the second question, for then, in trying to realise a mathematical tiling, they can ignore tilings known to be equivalent to ones that have already been realised. Mathematicians study symmetry using abstract algebraic structures such as symmetry groups. We can characterize the structural properties of a tiling by associating to it algebraic constructions called invariants. If two tilings are equivalent, their invariants are the same. So, an understanding of the algebraic invariants of a tiling leads to some answers to the first question. In this project, we seek to gain a better understanding of some of these invariants, how symmetries manifest in them, and how to compute them, so that we can make progress in classifying mathematical quasicrystals. |

Role | Start date | End date | Funding source |
---|---|---|---|

Lead | 01 Apr 2019 | 31 Mar 2023 | EPSRC Engineering and Physical Sciences Research Council |

Characterising aperiodically ordered structures and understanding their properties has been a major area of activity since the discovery of quasicrystals in the 1980s. There are three types of spectral quantities which are commonly studied in this context: (1) the diffraction measure, which is the Fourier transform of the autocorrelation measure and is directly related to the outcome of diffraction experiments; (2) the dynamical spectrum, which is related to the action of translations on the system; (3) the Schroedinger spectrum, which is the energy spectrum of a (discrete) Schroedinger operator describing the motion of electrons in an aperiodic system. While a good understanding of the relationshop between diffraction and dynamical spectra has recently emerged, the relation between these two spectra and those of the corresponding (discrete) Schroedinger operator has remained elusive. This project aims at investigating this relationship, building on recent results that point to a key role played by Lyapunov exponents of matrix product cocycles for the diffraction spectrum. |

The Ellis semigroup of bijective substitutions (2022)

Kellendonk, Johannes and Yassawi, Reem

Groups, Geometry and Dynamics ((Early access))

Automorphisms of automatic shifts (2021-05)

Müllner, Clemens and Yassawi, Reem

Ergodic Theory and Dynamical Systems, 41(5) (pp. 1530-1559)

Lucas congruences for the Apéry numbers modulo *p*^{2} (2021)

Yassawi, Reem; Rowland, Eric and Krattenthaler, Christian

Integers ((In press))

How to prove that a sequence is not automatic (2021)

Allouche, J.-P.; Shallit, J. and Yassawi, R.

Expositiones Mathematicae ((Early Access))

The Ellis semigroup of bijective substitutions (2021)

Kellendonk, Johannes and Yassawi, Reem

Groups, Geometry, and Dynamics, 16(1) (pp. 29-73)

Automaticity and Invariant Measures of Linear Cellular Automata (2020-12)

Rowland, Eric and Yassawi, Reem

Canadian Journal of Mathematics, 72(6) (pp. 1691-1726)

Recognizability for sequences of morphisms (2019-11)

Berthé, Valérie; Steiner, Wolfgang; Thuswaldner, Jórg and Yassawi, Reem

Ergodic Theory and Dynamical Systems, 39(11) (pp. 2896-2931)

Topological rigidity of linear cellular automaton shifts (2018-08)

Fokkink, Robbert and Yassawi, Reem

Indagationes Mathematicae, 29(4) (pp. 1105-1113)

Reversing and extended symmetries of shift spaces (2018-02)

Baake, Michael; Roberts, John A. G. and Yassawi, Reem

Discrete & Continuous Dynamical Systems - Series A, 38(2) (pp. 835-866)

Profinite automata (2017-04)

Rowland, Eric and Yassawi, Reem

Advances in Applied Mathematics, 85 (pp. 60-83)

*p*-adic asymptotic properties of constant-recursive sequences (2017-02)

Rowland, Eric and Yassawi, Reem

Indagationes Mathematicae, 28(1) (pp. 205-220)

Orders that yield homeomorphisms on Bratteli diagrams (2017)

Bezuglyi, Sergey and Yassawi, Reem

Dynamical Systems, 32(2) (pp. 249-282)

Bratteli diagrams where random orders are imperfect (2017)

Janssen, J.; Quas, A. and Yassawi, R.

Proceedings of the American Mathematical Society, 145(2) (pp. 721-735)

Computing automorphism groups of shifts using atypical equivalence classes (2016)

Coven, Ethan M.; Quas, Anthony and Yassawi, Reem

Discrete Analysis, 2016, Article 611(3)

A family of sand automata (2015-02-02)

Faulkner, Nicholas and Yassawi, Reem

Theoretical Computer Science, 565 (pp. 50-62)

A characterization of *p*-automatic sequences as columns of linear cellular automata (2015-02)

Rowland, Eric and Yassawi, Reem

Advances in Applied Mathematics, 63 (pp. 68-89)

Automatic congruences for diagonals of rational functions (2015)

Rowland, Eric and Yassawi, Reem

Journal de Théorie des Nombres de Bordeaux, 27(1) (pp. 245-288)

Perfect orderings on finite rank Bratteli diagrams (2014-02)

Yassawi, Reem; Bezugly, Sergey and Kwiatkowski, J.

Canadian Journal of Mathematics, 66(1) (pp. 57-101)

Embedding Bratteli-Vershik systems in cellular automata (2010-10)

Pivato, Marcus and Yassawi, Reem

Ergodic Theory and Dynamical Systems, 30(5) (pp. 1561-1572)

Embedding odometers in cellular automata (2009)

Coven, Ethan M. and Yassawi, Reem

Fundamenta Mathematicae, 206 (pp. 131-138)

Prevalence of odometers in cellular automata (2007-03)

Coven, Ethan M.; Pivato, Marcus and Yassawi, Reem

Proceedings of the American Mathematical Society, 135(3) (pp. 815-821)

Asymptotic randomization of subgroup shifts by linear cellular automata (2006-08)

Maass, Alejandro; Martínez, Servet; Pivato, Marcus and Yassawi, Reem

Ergodic Theory Dynam. Systems, 26(4) (pp. 1203-1224)

Asymptotic randomization of sofic shifts by linear cellular automata (2006-08)

Pivato, Marcus and Yassawi, Reem

Ergodic Theory and Dynamical Systems, 26(4) (pp. 1177-1201)

Limit measures for affine cellular automata II (2004-12)

Pivato, Marcus and Yassawi, Reem

Ergodic Theory and Dynamical Systems, 24(6) (pp. 1961-1980)

Multiple mixing and local rank group actions (2003-08)

Yassawi, Reem

Ergodic Theory and Dynamical Systems, 23(4) (pp. 1275-1304)

Limit measures for affine cellular automata (2002-08)

Pivato, Marcus and Yassawi, Reem

Ergodic Theory and Dynamical Systems, 22(4) (pp. 1269-1287)

Multiple Mixing and Rank One Group Actions (2000-04-01)

del Junco, Andrés and Yassawi, Reem

Canadian Journal of Mathematics, 52(2) (pp. 332-347)

Aperiodic Order Meets Number Theory: Origin and Structure of the Field (2021-02-14)

Baake, M.; Coons, M.; Grimm, U.; Roberts, J.A.G. and Yassawi, R.

In: Wood, David R.; de Gier, Jan; Praeger, Cheryl E. and Tao, Terence eds. *2019-20 MATRIX Annals*. MATRIX Book Series (4) (pp. 663-667)

ISBN : 978-3-030-62496-5 | Publisher : Springer | Published : Cham

Attractiveness of the Haar measure for linear cellular automata on Markov subgroups (2006)

Maass, Alejandro; Martínez, Servet; Pivato, Marcus and Yassawi, Reem

In: Denteneer, Dee; den Hollander, Frank and Verbitskiy, Evgeny eds. *Dynamics & Stochastics: Festschrift in honor of M. S. Keane*. The Institute of Mathematical Statistics Lecture Notes - Monograph Series (pp. 100-108)

ISBN : 0-940600-64-1 | Publisher : Institute of Mathematical Statistics | Published : Beachwood, Ohio, USA