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- Dr Ian Short

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.

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 SL_{2}-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.

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:

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

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, SL_{2}-tilings, continued fractions, dessins d'enfants, hyperbolic geometry, and a host of other mathematical concepts.

*The Farey graph*

Name | Type | Parent Unit |
---|---|---|

Pure Mathematics Research Group | Group | Faculty of Mathematics, Computing and Technology |

Pure Maths: Analysis & Geometry Research Group | Group | Faculty of Mathematics, Computing and Technology |

Stability of the Denjoy–Wolff theorem (2021)

Christodoulou, Argyrios and Short, Ian

Annales Academiæ Scientiarum Fennicæ ((In press))

Classifying SL_{2}-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|*b _{n}*) (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)