Dr Tony Samuel BSc, PhD

Dr Tony Samuel

School of Mathematics
Lecturer in Mathematics and Statistics

Contact details

School of Mathematics
University of Birmingham
B15 2TT

Tony Samuel is a Lecturer in Mathematics and Statistics at the University of Birmingham working within the Analysis, and the Dynamical Systems and Topology research groups. He has a number of publications in leading international mathematical journals covering topics such as aperiodic order, Diophantine approximations, dynamical systems, non-commutative geometry and potential theory.  He regularly collaborates with mathematicians around the world, with co-authors from China, Germany and the US, and has organised a number of conferences, workshops and graduate schools in Germany (Bremen and Lübeck), Sweden (Djursholm), the UK (Birmingham) and the US (San Luis Obispo, CA). Tony has also supervised a number of graduate and undergraduates.  If you are interested in working with him, he would be interested in hearing from you.

For more information, please visit Tony's School of Mathematics homepage.


  • PhD – Pure Mathematics – University of St Andrews, Scotland (2010)
  • BSc Hons. – Mathematics – University of St Andrews, Scotland (2005)


Tony graduated from the University of St Andrews in 2005, and went on to complete a PhD in Pure Mathematics under the supervision of Prof. K. Falconer (University of St Andrews) and Prof. B. O. Stratmann (Universität Bremen), where he studied ways in which noncommutative geometry can be used to understand fractal sets. For his doctoral thesis, Tony was awarded an EPSRC Doctoral Prize, which funded a one year Postdoctoral Fellowship at the University of St Andrews. Following this, in 2011 Tony worked under the guidance of Prof. M. Barnsley (Austrilian National University) and Prof. A. Rennie (University of Wollongong) on ARC Discovery Project: DP0984353, where they continued the work initiated in Tony's doctoral thesis.

Subsequently, Tony has held posts at the University of Bremen as a Scientific Researcher (Wissenschaftlicher Mitarbeiter) and California Polytechnic State University as an Assistant Professor of Pure Mathematics. Further, in 2017 he spent a semester at Institut Mittag-Leffler (Sweden) where he took part in the research program Fractal Geometry and Dynamics led by K. Falconer, M. Järvenpää, A. Kupiainen, F. Ledrappier and P. Mattila.


Tony has coordinated/taught over 20 lecture courses, at all levels, ranging from first year courses in mathematics, through to advanced topics such as fractal geometry, measure theory and topology.  He has led a number of undergraduate and graduate seminars, and has advised several MSc and BSc theses. Tony is open to new teaching methods and has successfully used techniques such as enquiry based learning, flipped classroom and research based learning in his lectures.

In his role as an advisor, Tony has arranged for undergraduates and graduates to take part in various overseas research programs and conferences. Further, he has been involved in outreach projects including: organising public lectures, mathematical demonstrations and exhibits at university open days, as well as being a member of student welfare programs to promote inclusivity and diversity.

Postgraduate supervision

Tony is keen to supervise postdocs and research students in a number of areas of analysis, in particular, aperiodic order, Diophantine approximations, dynamical systems, ergodic theory and geometry. Current and previous doctoral students include the following.

  • K. Sender: Diffusion on irregular sets. Jointly supervised with M. Kesseböhmer.
  • A. Mosbach: Finite and infinite rotation sequences and beyond. Jointly supervised with M. Kesseböhmer.
  • M. Steffens, PhD  (2018): Regularity of aperiodic subshifts and connections to intermediate β-transformations. Jointly supervised with M. Kesseböhmer.
  • H. Weyer, PhD (2018): A study on measure-geometric Laplacians on the real line. Jointly supervised with M. Kesseböhmer.


Tony’s research interests include aperiodic order, Diophantine approximations, dynamical systems (symbolic and measure preserving), ergodic theory (finite and infinite), geometry (fractal, hyperbolic and noncommutative) and potential theory. Recently, his focus has been on applications of ergodic theory, noncommutative geometry and potential theory to fractals and quasicrystals. He is also very interested in linking these topics to other areas of mathematics, such as, geometric group theory, geometric measure theory and renewal theory.

Other activities

  • Member of the London Mathematical Society (Jan. 2014 – Present)
  • Member of the American Mathematical Society (Nov. 2006 – Dec. 2018)
  • Member of the International Group for the Psychology of Mathematics Education (Jan. 2017 – Dec. 2017)
  • Member of the DFG Scientific Network: Skew Product Dynamics and Multifractal Analysis (Jan. 2012 – Dec. 2016)


Books (Editor)

  • Horizons of Fractal Geometry and Complex Dimensions. Contemp. Math. (Expected publication year 2019). Editors: R. G. Niemeyer, E. P. J. Pearse, J. A. Rock and T. Samuel.
  • Diffusion on Fractals and Non-linear Dynamics: Discrete Contin. Dyn. Syst. Ser. S 10(2), 161-394 (2017). Editors: K. Falk, T. Jäger, M. Kesseböhmer, J. Rademacher and T. Samuel.


  • M. Kesseböhmer, A. Mosbach, T. Samuel and M. Steffens. Diffraction of return time measures: To appear in J. Stat. Phys. 11 pages (2018).
  • B. Li, T. Sahlsten, T. Samuel and W. Steiner. Denseness of intermediate β-shifts of finite type: Accepted for publication in Proc. Amer. Math. Soc. 10 pages (2018).
  • M. Kesseböhmer, T. Samuel and K. Sender. The Sierpiński gasket as the Martin boundary of a non-isotropic Markov chain: Accepted for publication in J. Fractal Geom. 21 pages (2018).
  • M. Gröger, M. Kesseböhmer, A. Mosbach, T. Samuel and M. Steffens. A classification of aperiodic order via spectral metrics and Jarnìk sets: Accepted for publication in Ergod. Dyn. Sys. 38 pages (2018).
  • M. Kesseböhmer, T. Samuel and H. Weyer. Measure-geometric Laplacians for discrete distributions: Accepted for publication in Contemp. Math. 8 pages (2017).
  • F. Dreher, M. Kesseböhmer, A. Mosbach, T. Samuel and M. Steffens. Regularity of aperiodic minimal subshifts: Bull. Math. 8(3) 413-434 (2018). 
  • M. Kesseböhmer, T. Samuel and H. Weyer. A note on measure-geometric Laplacians: Monatsh. Math. 181(3), 643-655 (2016).
  • B. Li, T. Sahlsten and T. Samuel. Intermediate beta-shifts of finite type: Discrete Contin. Dyn. Syst. 36(1), 323-344 (2016).
  • J. Kautzsch, M. Kesseböhmer and T. Samuel. On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps: Ann. Henri Poincaré 17(9), 2585-2621 (2016).
  • J. Kautzsch, M. Kesseböhmer, T. Samuel and B. O. Stratmann. On the asymptotics of the α-Farey transfer operator: Nonlinearity 28, 143-166 (2015).
  • F. Dreher and T. Samuel. Continuous images of Cantor's ternary set: Amer. Math. Monthly 121(7), 640-643 (2014). Translated into Mandarin: Shu Xue Yi Lin 35(4), 381-384 (2016).
  • T. Samuel, N. Snigireva and A. Vince. Embedding the symbolic dynamics of Lorenz maps: Math. Proc. Camb. Phil. Soc. 156(3), 505-519 (2014).
  • M. Kesseböhmer and T. Samuel. Spectral metric spaces for Gibbs measures: J. Funct. Anal. 31, 1801-1828 (2013).
  • T. Samuel. A simple proof of Vitali's theorem for signed measures: Amer. Math. Monthly 120(7), 654-660 (2013).
  • K. Falconer and T. Samuel. Dixmier traces and coarse multifractal analysis: Ergod. Dyn. Sys. 31, 369-381 (2011).