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

I am currently advertising a PhD project to study the "Iteration of Quasiregular Mappings".

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.
My work in this area can be roughly divided into two topics: firstly, deficient values and deficient functions of certain classes of meromorphic functions. Here a value is called *deficient* if a function takes that value less often than it might be expected to (i.e. less often than the function takes most other values).
The second topic is connected to real functions; that is, entire or meromorphic functions that are real on the real axis. This has included problems related to Wiman's conjecture which dates back to 1911, but was resolved in 2002.

- (with D. J. Sixsmith) The size and topology of quasi-Fatou components of quasiregular maps, submitted. arXiv:1601.03308.
- Slow escaping points of quasiregular mappings, submitted. arXiv:1511.01799.
- (with D. J. Sixsmith) Periodic domains of quasiregular maps, submitted. arXiv:1509.06723.
- (with D. J. Sixsmith) Hollow quasi-Fatou components of quasiregular maps, submitted. arXiv:1505.08114.
- (with P. J. Rippon and G. M. Stallard) Baker's conjecture for functions with real zeros, submitted. arXiv:1502.02526.
- (with A. Fletcher) Superattracting fixed points of quasiregular mappings ---
*Ergodic Theory Dynam. Systems*, published online, to appear in print. 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. - (with A. Fletcher) Iteration of quasiregular tangent functions in three dimensions ---
*Conform. Geom. Dyn.***16**(2012), 1-21. arXiv:1112.3589. - Wandering domains in quasiregular dynamics ---
*Proc. Amer. Math. Soc.***141**(2013), 1385-1392. arXiv:1101.1483. - (with A. Fletcher) Julia sets of uniformly quasiregular mappings are uniformly perfect ---
*Math. Proc. Cambridge Philos. Soc.***151**(2011), 541-550. arXiv:1012.1378. - Non-real zeroes of real entire derivatives ---
*J. Anal. Math.***117**(2012), 87-118. DOI:10.1007/s11854-012-0015-5. - (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 "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 organised the One Day Function Theory Meeting that took place on 2 September 2013 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 C54 in the Mathematics Building.Last updated: March 2015