I am a Lecturer in Analysis within the School of Mathematical Sciences at the University of Nottingham. My contact details are given below.

My research has its origins in complex analysis. I study analytic and meromorphic functions on the complex plane and quasiregular mappings of n-dimensional real space. I am interested in the behaviour under iteration of such functions, and also their value distribution.

Quasiregular mappings are a natural generalisation to higher dimensions of analytic functions on the plane. Roughly, a quasiregular map is one which sends infinitesimal spheres to infinitesimal ellipsoids with bounded eccentricity. The iteration of complex analytic functions has been a very active and successful field in recent years and the study of 'quasiregular dynamics' is now emerging as an exciting new branch, lying between the well-studied analytic case (where many powerful tools are available) and general dynamics in several real variables, which is much less well-understood.

The considerable flexibility of quasiregular mappings means that there are fundamental differences to the analytic case. Nonetheless, research is showing, perhaps surprisingly, that some features of complex dynamics do persist in the quasiregular setting.

The image on the left shows an example of a quasiregular escaping set, the set of points whose images tend to infinity under iteration. The boundary of an escaping set is related to a function's Julia set. The Julia set is central to complex dynamics, but identifying the correct analogue for Julia sets in quasiregular dynamics is not straightforward. This image was produced by A. Fletcher and D. Goodman. See more pics here, and also this gallery related to a quasiregular version of the tangent function.

My PhD thesis focussed on the value distribution of meromorphic functions. Value distribution theory explores the relationship between the rate of growth of a function and the frequency with which the function takes different values.

- (with A. Fletcher) Normal families and quasiregular mappings ---
*Proc. Edinb. Math. Soc. (2)*, FirstView (2023), to appear in print. arXiv:2201.08921. - (with D. J. Sixsmith) Which sequences are orbits? ---
*Anal. Math. Phys.***11**, paper no. 53 (2021). arxiv:1907.11006. - (with P. J. Rippon and G. M. Stallard) Iterating the minimum modulus: functions of order half, minimal type ---
*Comput. Methods Funct. Theory***21**(2021), 653-670. arXiv:2102.02158. - (with P. J. Rippon and G. M. Stallard) Eremenko's conjecture for functions with real zeros: the role of the minimum modulus ---
*Int. Math. Res. Not. IMRN***18**(2021), 13946-13974. arxiv:1810.07814. - (with D. J. Sixsmith) The bungee set in quasiregular dynamics ---
*Bull. Lond. Math. Soc.***51**(2019), 120-128. arXiv:1803.01716. - (with D. J. Sixsmith) The dynamics of quasiregular maps of punctured space ---
*Indiana Univ. Math. J.***68**(2019), 323-352. arXiv:1607.06649. Published pdf here. - (with P. J. Rippon and G. M. Stallard) Baker's conjecture for functions with real zeros ---
*Proc. Lond. Math. Soc.***117**(2018), 100-124. arXiv:1502.02526. - (with D. J. Sixsmith) Periodic domains of quasiregular maps ---
*Ergodic Theory Dynam. Systems***38**(2018), 2321-2344. arXiv:1509.06723. - (with D. J. Sixsmith) Hollow quasi-Fatou components of quasiregular maps ---
*Math. Proc. Cambridge Philos. Soc.***162**(2017), 561-574. arXiv:1505.08114. - (with D. J. Sixsmith) The size and topology of quasi-Fatou components of quasiregular maps ---
*Proc. Amer. Math. Soc.***145**(2017), 749-763. arXiv:1601.03308. - Slow escaping points of quasiregular mappings ---
*Math. Z.***284**(2016), 1053-1071. arXiv:1511.01799. - (with A. Fletcher) Superattracting fixed points of quasiregular mappings ---
*Ergodic Theory Dynam. Systems***36**(2016), 781-793. arXiv:1404.2778. - (with W. Bergweiler and A. Fletcher) The Julia set and the fast escaping set of a quasiregular mapping ---
*Comput. Methods Funct. Theory***14**(2014), 209-218. arXiv:1309.4601. - (with W. Bergweiler) Foundations for an iteration theory of entire quasiregular maps ---
*Israel J. Math.***201**(2014), 147-184. arXiv:1210.3972. - (with A. Fletcher) Chaotic dynamics of a quasiregular sine mapping ---
*J. Difference Equ. Appl.***19**(2013), 1353-1360. arXiv:1208.3585. - Wandering domains in quasiregular dynamics ---
*Proc. Amer. Math. Soc.***141**(2013), 1385-1392. arXiv:1101.1483. - (with A. Fletcher) Iteration of quasiregular tangent functions in three dimensions ---
*Conform. Geom. Dyn.***16**(2012), 1-21. arXiv:1112.3589. - Non-real zeroes of real entire derivatives ---
*J. Anal. Math.***117**(2012), 87-118. DOI:10.1007/s11854-012-0015-5. - (with A. Fletcher) Julia sets of uniformly quasiregular mappings are uniformly perfect ---
*Math. Proc. Cambridge Philos. Soc.***151**(2011), 541-550. arXiv:1012.1378. - (with A. Fletcher) Quasiregular dynamics on the
*n*-sphere ---*Ergodic Theory Dynam. Systems***31**(2011), 23-31. arXiv:0909.0217. - Real meromorphic functions and a result of Hinkkanen and Rossi ---
*Illinois J. Math.***53**(2009), 605-622. - Deficiencies of certain classes of meromorphic functions ---
*Ann. Acad. Sci. Fenn. Math.***34**(2009), 157-171. - Rational deficient functions of derivatives of mappings in the classes
*S*and*B*---*Comput. Methods Funct. Theory***9**(2009), 239-253.

Slides from "Iterating the minimum modulus", a talk given in the CAvid Complex Analysis seminar series, March 2022.

Slides from "Orbits and bungee sets", a talk given online for On geometric complexity of Julia sets - III, Bedlewo, Poland, September 2021.

Slides from "The iterated minimum modulus and Eremenko's conjecture", a talk given at Dynamics, Equations and Applications (DEA2019), Krakow, September 2019.

Slides from "The quasi-Fatou set in quasiregular dynamics", a talk given at an ICMS Workshop celebrating Walter Bergweiler's 60th birthday, Edinburgh, July 2018.

Slides from "Julia sets in higher dimensions", a talk given at the LMS Graduate Student Meeting, London, June 2017.

Slides from "Fast escape using normal families", a talk given at the Postgraduate Conference on Complex Dynamics, London, March 2015.

Slides from "Superattracting fixed points of quasiregular functions", a talk given in Bedlewo, Poland, July 2014.

Slides from "The Julia set in quasiregular dynamics", a talk given at an ICMS Workshop, Edinburgh, May 2013.

Slides from "Iteration of quasiregular analogues of trigonometric functions", a talk given at the One Day Function Theory Meeting, London, September 2012.

Slides from "Iteration of quasiregular tangent functions", a talk given in Bedlewo, Poland, April 2012.

Slides from "Real meromorphic functions", a talk given at the CMFT conference, Ankara, June 2009.

I co-ordinate the LMS Scheme 3 Network in *Holomorphic Dynamics*. This does not currently have a website, but please contact me for any information. We hold meetings roughly twice a year, with our next one planned for spring / summer 2024 at the University of Liverpool.

I organised the 2013 and 2018 iterations of the One Day Function Theory Meeting at De Morgan House, London.

A book review of "Early days in complex dynamics" by Alexander, Iavernaro and Rosa. This review appeared in the Bulletin of the LMS.

My PhD thesis written under the supervision of Prof Jim Langley.

My Part III essay on Automorphic Functions written under the supervision of Prof Alan Beardon.

I highly recommend Jim Langley's postgraduate notes as an excellent introduction to many areas of complex analysis.

*Dr Daniel Nicks*

School of Mathematical Sciences, University of Nottingham, NG7 2RD.

Tel: +44 (0)115 9514965

Email: dan.nicks "at" nottingham.ac.uk

My office is B42 in the Mathematics Building.Last updated: December 2023