I am currently a DAAD PRIME Fellow at the Open University, supported by the German Academic Exchange Service (DAAD). I did my PhD with Prof. Michael Baake at Universität Bielefeld in Bielefeld, Germany, which I completed in June 2019. My PhD Thesis is on Lyapunov exponents and related criteria which characterise the spectral type of aperiodic tilings and subshifts.
My main research interests are aspects of aperiodic order, in particular, dynamical systems generated by sequences, tilings, and measures satisfying some form of hierarchy or self-similarity. Some prevalent themes in my research, together with recent and ongoing work, are listed below.
- Lyapunov exponents in spectral theory
- Together with Michael Baake, Franz Gaehler and Uwe Grimm, we have developed a criterion which rules out the presence of absolutely continuous spectrum in inflation tilings both in one and in higher dimensions, which can be succinctly written as a bound condition on the Lyapunov exponent of the Fourier cocycle. The general framework is presented in https://arxiv.org/abs/1805.09650 (with M. Baake and F. Gaehler, Commun.Math.Phys) and specific classes of examples are treated in https://arxiv.org/abs/1706.00451 (J.Math.Phys.) and https://arxiv.org/abs/1709.09083 (with M. Baake and U. Grimm, Lett.Math.Phys.).
- We establish a connection between Lyapunov exponents of certain substitutions and logarithmic Mahler measures in https://arxiv.org/abs/1711.02492 (with M. Baake and M. Coons, From Analysis to Visualization, Springer Conf. Ser.).
- Dynamical properties of random substitutions
- Using generalised Zeckendorf representations of natural numbers, we demonstrate in https://arxiv.org/abs/1912.01573 (with E. Miro, D. Rust and G. Tadeo, Tsukuba. J. Math.) that several examples of random substitution subshifts are not topologically mixing. We generalise this result and explore the topological entropies of a larger class in https://arxiv.org/abs/2103.04866 (with E. Miro and G. Escolano, to appear: Indag. Math.).
- Algebraic invariants of higher-dimensional subshifts
- With Alvaro Bustos and Daniel Luz, we fully characterise the groups which appear as groups of extended symmetries of higher-dimensional bijective substitution subshifts, and we provide an algorithm how to compute the group given any example. The work https://arxiv.org/abs/2101.10988 (to appear, Discrete Comput. Geom.) also contains realisation results on how to build systems with preset automorphism and extended symmetry groups.
- Measures arising from regular sequences
- With Michael Coons and James Evans, we develop a spectral theory for (possibly unbounded) regular sequences, which are generalisations of automatic sequences by developing a framework in how one can associate a probability measure to them and (possibly) to the R-vector space they generate. In https://arxiv.org/abs/2009.01402 (to appear, Documenta Math.), some criteria are provided which guarantee the existence of the limit measure. In https://arxiv.org/abs/2108.05007 (with M. Coons, J. Evans, Z. Groth), we prove that the distribution function satisfies some self-affine properties and provide conditions for the limit measure to be singular continuous.
- Group-theoretic constructions in spectral theory
- In https://arxiv.org/abs/2108.08642, with Natalie Priebe Frank, we generalise Rudin--Shapiro, geometrically, algebraically and spectrally by constructing substitution subshifts whose periodic skeleton derives from digit tilings of Z^d. We call them spin qubit substitutions, alluding to the underlying digit structure and the abelian group structure of spins which allows access to the spectral measures.
- We consider in https://arxiv.org/abs/2108.01762 (with Dan Rust and Jamie Walton) substitutions on compact abelian groups, and prove several criteria for the diffraction spectrum to be pure point or purely singular continuous, again relying on the underlying group structure.
- Substitutions on infinite alphabets
- In https://arxiv.org/abs/2204.07516, together with Dan Rust and Jamie Walton, we contribute to the systematic treatment of substitutions of compact alphabets, and in particular explore their topological, ergodic-theoretic and operator-theoretic properties. We give sufficient conditions for unique ergodicity of the subshift and for the existence of a natural length function for a given substitution.
- In ongoing work with Alexey Garber and Dirk Frettloeh, we consider a class of substitutions on specific compactifications of the naturals and demonstrate how they exhibit properties which are not possible for substitutions on finite alphabets.
- Arithmetic progressions in automatic sequences
- In ongoing work with Ibai Aedo, Yasushi Nagai and Petra Staynova, we aim to quantify the asymptotic behavior of long arithmetic progressions in automatic sequences, in light of van der Waarden's result for arbitrary colourings of the integers.
Before moving to the Open University, I have served as a tutor for a variety of courses (bachelors and masters level) while I was in Bielefeld from 2015 to 2021. These include partial differential equations, algebra, linear algebra for physicists, group and representation theory for physicists, Fourier analysis for life sciences, complex analysis, probability theory and statistics, and discrete mathematics.
Before moving to Germany for my PhD, I served as an instructor at the Mathematics department of Ateneo de Manila University in the Philippines, where I handled service courses in college algebra and calculus for business.
I have been awarded the Dissertationspreis for the best PhD thesis in the Faculty of Mathematics at Bielefeld University in 2019. This was selected among all Phd theses who were awarded the best grade (summa cum laude) during that year. In 2020, I was awarded a DAAD Postdoctoral Researchers International Mobility Experience (PRIME) fellowship, which includes an 12-month research stay at the Open University and a 6-month return phase at Bielefeld University. This fellowship is funded by the German Federal Ministry of Education and Research (BMBF) and the European Union (EU/FP7/Marie Curie Actions/COFUND).
I am currently working on projects together with several colleagues from Japan, the Philippines, the USA, Germany, and Australia.
Spectral theory of spin substitutions (2022)
Frank, Natalie Priebe and Manibo, Neil
Discrete and Continuous Dynamical Systems, 42(11) (pp. 5399-5435)