Dan Nicks

Photo

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

Research

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.

A slice through the escaping set of a Zorich map. Pic courtesy of Alastair Fletcher and Dan Goodman. 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.


Publications and pre-prints

Other

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.

The slides from a talk on "The Julia set in quasiregular dynamics" given at an ICMS Workshop, Edinburgh, May 2013.
The slides from a talk on "Iteration of quasiregular analogues of trigonometric functions" given at the One Day Function Theory Meeting, London, September 2012.
The slides from a talk on "Iteration of quasiregular tangent functions" given in Bedlewo, Poland, April 2012.
The slides from a talk on "Real meromorphic functions" given at the CMFT conference, Ankara, June 2009.

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.


Contact Details

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: April 2014