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. 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.
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 am organising the One Day Function Theory Meeting that will take place on 3 September 2018 at De Morgan House, London. I also organised the 2013 iteration of this meeting.
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.ukMy office is B42 in the Mathematics Building.