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Dr Ian Short

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Profile summary

Professional biography

I am the Director of Research in Mathematics and Statistics and a Senior Lecturer in Analysis. I joined the Open University in 2010 having previously worked at the University of Cambridge and the National University of Ireland Maynooth. Before that I was PhD student at the University of Cambridge.

Research interests

My research interests include geometry and dynamics, complex analysis, continued fractions, and group theory. Here are some recent projects.

  • Classifying SL2-tilings https://arxiv.org/abs/1904.11900. This work describes a new, unifying approach to classifying SL2-tilings using the geometric, numeric, and combinatorial properties of the Farey graph. To be published in the Transactions of the American Mathematical Society.
  • Semigroups of isometries of the hyperbolic plane https://arxiv.org/abs/1609.00576 (joint with Matthew Jacques). This paper relates nonautonomous dynamical systems of hyperbolic isometries with semigroups of hyperbolic isometries, and develops the theory of such semigroups. To be published in International Mathematics Research Notices.
  • Necessary and sufficient conditions for convergence of continued fractions http://arxiv.org/abs/2102.10173 (with Margaret Stanier). We describe a simple algorithm for determining whether an integer continued fraction converges, inspired by the geometry of the Farey graph. To be published in the Proceedings of the American Mathematical Society
  • Nilpotent covers of symmetric groups https://arxiv.org/abs/2005.13869 (with Nick Gill and Kimei Arphaxad Ngwava). We prove that the symmetric group on n letters has a unique minimal cover by maximal normal subgroups. This cover has striking properties; for example, the number of conjugacy classes in the cover is equal to the number of partitions of n into distinct positive integers.

Teaching interests

I am chair of the undergraduate module M337 Complex analysis and the MSc module M829 Analytic number theory II

I chaired the rewrite of M337, completed in 2018. The revised books for M337 have approximately 1150 pages, well over 1000 figures, and a wealth of exercises and solutions. Here are the four beautifully illustrated module texts and the module handbook:

M337 texts

Before writing M337, I was an author of the modules MST124 Essential mathematics 1 and MST125 Essential mathematics 2.

Impact and engagement

I lead the Navigating by numbers outreach programme. The purpose of the programme is to communicate an attractive network of geometric and numeric ideas centred around the Farey graph. This involves Coxeter's frieze patters, SL2-tilings, continued fractions, dessins d'enfants, hyperbolic geometry, and a host of other mathematical concepts.

Farey graph
The Farey graph

Research groups

NameTypeParent Unit
Pure Mathematics Research GroupGroupFaculty of Mathematics, Computing and Technology
Pure Maths: Analysis & Geometry Research GroupGroupFaculty of Mathematics, Computing and Technology

Publications

Stability of the Denjoy–Wolff theorem (2021)
Christodoulou, Argyrios and Short, Ian
Annales Academiæ Scientiarum Fennicæ ((In press))


Classifying SL2-tilings (2021)
Short, Ian
Transactions of the American Mathematical Society ((In Press))


Semigroups of isometries of the hyperbolic plane (2021)
Jacques, Matthew and Short, Ian
International Mathematics Research Notices, Article rnaa291 ((Early Access))


Necessary and sufficient conditions for convergence of integer continued fractions (2021)
Short, Ian and Stanier, Margaret Rosemary
Proceedings of the American Mathematical Society ((In press))


Normal Families of Möbius Maps (2020-03-05)
Beardon, Alan; Minda, David and Short, Ian
Computational Methods and Function Theory, 20 (pp. 523-538)


Repeated compositions of Möbius transformations (2020)
Jacques, Matthew and Short, Ian
Ergodic Theory and Dynamical Systems, 40(2) (pp. 437-452)


Escaping sets of continuous functions (2019-03)
Short, Ian and Sixsmith, David J.
Journal d'Analyse Mathematique, 137(2) (pp. 875-869)


A hyperbolic-distance inequality for holomorphic maps (2019)
Short, Ian and Christodoulou, Argyrios
Annales Academiæ Scientiarum Fennicæ, Mathematica, 44 (pp. 293-300)


Substitution-based structures with absolutely continuous spectrum (2018-08-31)
Chan, Lax; Grimm, Uwe and Short, Ian
Indagationes Mathematicae, 29(4) (pp. 1072-1086)


Geodesic Rosen Continued Fractions (2016-12-30)
Short, Ian and Walker, Mairi
The Quarterly Journal of Mathematics, 67(4) (pp. 519-549)


The parabola theorem on continued fractions (2016-04-11)
Short, Ian
Computational Methods and Function Theory, 16(4) (pp. 653-675)


A geometric representation of continued fractions (2014-05)
Beardon, Alan F. and Short, Ian
American Mathematical Monthly, 121(5) (pp. 391-402)


Maximal buttonings of trees (2014)
Short, Ian
Discussiones Mathematicae Graph Theory, 34(2) (pp. 415-420)


On the product decomposition conjecture for finite simple groups (2013-11)
Gill, Nick; Pyber, László; Short, Ian and Szabó, Endre
Groups, Geometry, and Dynamics, 7(4) (pp. 867-882)


Conformal automorphisms of countably connected regions (2013-01-09)
Short, Ian
Conformal Geometry and Dynamics, 17 (pp. 1-5)


Conjugacy in Thompson's group $F$ (2013)
Short, Ian and Gill, Nick
Proceedings of the American Mathematical Society, 141 (pp. 1529-1538)


Hausdorff dimension of sets of divergence arising from continued fractions (2012-04)
Short, Ian
Proceedings of the American Mathematical Society, 140(4) (pp. 1371-1385)


Flowability of plane homeomorphisms (2012)
O'Farrell, Anthony; Le Roux, Frédéric; Roginskaya, Maria and Short, Ian
Annales de l'Institut Fourier, 62(2) (pp. 619-639)


Geodesic continued fractions (2012)
Beardon, A. F.; Hockman, M. and Short, I.
Michigan Mathematical Journal, 61(1) (pp. 133-150)


Visualizing uncertainty about the future (2011-09-09)
Spiegelhalter, David; Pearson, Mike and Short, Ian
Science, 333(6048) (pp. 1393-1400)


Rigidity of configurations of balls and points in the N-sphere (2011-06-29)
Crane, Edward and Short, Ian
The Quarterly Journal of Mathematics, 62(2) (pp. 351-362)


Ford circles, continued fractions, and rational approximation (2011-02)
Short, Ian
American Mathematical Monthly, 118(2) (pp. 130-135)


The Seidel, Stern, Stolz and Van Vleck Theorems on continued fractions (2010-06)
Beardon, Alan F. and Short, Ian
Bulletin of the London Mathematical Society, 42(3) (pp. 457-466)


Norms of Möbius maps (2010-06)
Beardon, Alan F. and Short, Ian
Bulletin of the London Mathematical Society, 422(3) (pp. 499-505)


Reversible maps and composites of involutions in groups of piecewise linear homeomorphisms of the real line (2010-03)
Gill, Nick and Short, Ian
Aequationes Mathematicae, 79(1-2) (pp. 23-37)


Reversibility in the group of homeomorphisms of the circle (2009-10)
Gill, Nick; O'Farrell, Anthony G. and Short, Ian
Bulletin of the London Mathematical Society, 41(5) (pp. 885-897)


Conjugacy classification of quaternionic Möbius transformations (2009)
Parker, John R. and Short, Ian
Computational Methods and Function Theory, 9(1) (pp. 13-25)


Reversibility in the diffeomorphism group of the real line (2009)
O'Farrell, Anthony and Short, Ian
Publicacions Matemàtiques, 53(2) (pp. 401-405)


Reversible maps in isometry groups of spherical, Euclidean and hyperbolic space (2008-09-18)
Short, Ian
Mathematical Proceedings of the Royal Irish Academy, 108A(1) (pp. 33-46)


How much money do you need? (2008-01)
Short, Ian
Mathematical Spectrum, 40(2) (pp. 67-72)


Conical limit sets and continued fractions (2007-10-31)
Crane, Edward and Short, Ian
Conformal Geometry and Dynamics, 11 (pp. 224-249)


Conformal symmetries of regions (2007-07)
Beardon, Alan F. and Short, Ian
Irish Mathematical Society Bulletin, 59(1) (pp. 49-60)


Reversible maps in the group of quaternionic Möbius transformations (2007)
Lávička, Roman; O'Farrell, Anthony G. and Short, Ian
Mathematical Proceedings of the Cambridge Philosophical Society, 143(1) (pp. 57-69)


Van Vleck's theorem on continued fractions (2007)
Beardon, Alan F. and Short, Ian
Computational Methods and Function Theory, 7(1) (pp. 185-203)


Magic letter groups (2007)
Short, Ian and Pearson, Mike
Mathematical Gazette, 91(522) (pp. 493-499)


Hyperbolic geometry and the Hillam-Thron theorem (2006-04-07)
Short, Ian
Geometriae Dedicata, 119(1) (pp. 91-104)


The hyperbolic geometry of continued fractions K(1|bn) (2006)
Short, Ian
Annales Academiae Scientiarum Fennicae Mathematica, 31(2) (pp. 315-327)


A counterexample to a continued fraction conjecture (2006)
Short, Ian
Proceedings of the Edinburgh Mathematical Society. Series II, 49(3) (pp. 735-737)


Reversibility in Dynamics and Group Theory (2015-05-31)
O'Farrell, Anthony G. and Short, Ian
London Mathematical Society Lecture Note Series
ISBN : 9781107442887 | Publisher : Cambridge University Press | Published : Cambridge


Even-integer continued fractions and the Farey tree (2014)
Short, Ian and Walker, Mairi
In : SIGMAP 2014 (7 Jul 2014-11 July 2014, West Malvern, UK)