I am a postgraduate student at the University of Nottingham, studying for a PhD in maths. My chosen field is Nevanlinna Theory, a part of Complex Analysis which allows for the analysis of meromorphic functions in the plane. Analysis of such functions using classical methods, such as the Maximum Modulus Principle, break down for meromorphic functions due to their poles. Nevanlinna Theory allows us to bypass this problem.

 

Published work
Title Date Journal Abstract
Pairs of non-homogeneous linear differential polynomials 2010 Computational Methods and Function Theory
11, no. 1, 2011, pp283-300		 In [8], Langley proved a result concerning the zeros of pairs of (possibly non-homogeneous) linear differential polynomials in a meromorphic function. We generalise this result by relaxing Langley's assumption on the frequency of the zeros (counting multiplicity), and further prove some results based on restricting the order of the differential operators.
Non-linear homogeneous differential polynomials 2011 Computational Methods and Function Theory
12, no. 1, 2012, pp145-150		 We apply lemmas of Mues and Steinmetz from [4] to non-linear homogeneous differential polynomials in the meromorphic function f and f(k) with coefficients which are O(log r) + o(T(r,f)) in order to find sufficient conditions for f to be of the form ReP where R is a rational function and P is a polynomial.

 

 

Awaiting publication
Title Date Status Abstract
Integer points of meromorphic functions 2011 Submitted to Proceedings of the Edinburgh Mathematical Society Working from a half-plane result of Fletcher and Langley [2], we show that if f is an integer-valued function on some subset of the natural numbers of positive lower density and is meromorphic of sufficiently small exponential type in the plane, then f is a polynomial.
A theorem on homogeneous differential polynomials 2011 Accepted by Results in Mathematics We substantially strengthen an unpublished result of Whitehead from his PhD thesis [8] using a refinement of his techniques.

 

 

Other Work
Title Date Explanation Abstract
Hilbert's Problems 2007 My third year project. This report is an investigation into Hilbert's Problems, a set of twenty-three questions posed by David Hilbert in 1900. In particular, the report focuses on the first and third problems - those of the Continuum Hypothesis, following Cohen's original proof of its independence; and the Equidecomposability of Polyhedra, following Hadwiger's proof. It also looks at sets of problems for the twenty-first century.
Nevanlinna Theory - Interim Report 2007 The end-of-semester report on my dissertation. My dissertation is on the topic of Nevanlinna Theory, a powerful tool in Complex Analysis. My aim is to provide an overview of the basic theorems and results of the subject, and then investigate where it has been used in research.
Nevanlinna Theory 2008 The end result of my dissertation. This dissertation is on the topic of Nevanlinna Theory, a powerful tool in Complex Analysis. In this report, I begin by studying how Nevanlinna Theory is derived, and continue by showing how its results and methods can be used to solve some interesting problems which are simple to state, but often not to resolve. I end by looking at how Nevanlinna Theory has been used by other authors in recent research.
Year 1 Report - Pairs of Non-Homogeneous Linear Differential Polynomials 2010 The report I produced at the end of the first year of my PhD. In [8], Langley proved a result concerning the zeros of pairs of (possibly non-homogeneous) linear differential polynomials in a meromorphic function. We generalise this result by relaxing Langley's assumption on the frequency of the zeros (counting multiplicity), and further prove some results based on restricting the order of the differential operators.
Nevanlinna Theory 2010 PowerPoint slides for my first year PhD presentation. We first look at the definitions of Nevanlinna Theory, and some basic results. We then look at a result by Langley concerning pairs of linear differential polynomials, and generalise it.
Pairs of non-homogeneous linear differential polynomials 2011 A poster I presented at the first Frontiers of Nevanlinna Theory conference at UCL. In [4], Langley proved a result concerning the zeros of pairs of (possibly non-homogeneous) linear differential polynomials in a meromorphic function. We generalise this result by relaxing Langley's assumption on the frequency of the zeros (counting multiplicity), and further prove some results based on restricting the order of the differential operators.
2nd year Research Report 2011 The report I produced at the end of the second year of my PhD. We first present an introduction to Nevanlinna Theory, covering the basic definitions and results. We then perform a brief literature review and state a research and thesis plan for the remainder of my time. Finally, we present a draft thesis chapter, entitled Non-linear homogeneous differential polynomials.
Integer points of meromorphic functions 2011 Slides to accompany a talk I gave at the 2011 One Day Function Theory Meeting. An integer-valued function is a function which takes integer values on some subset of the natural numbers. We review the history of research on integer-valued functions, then present a result on when an integer-valued meromorphic function must be a polynomial.

 

 

Conferences I have attended
Title Date Location Organiser Notes
One Day Function Theory Meeting and Three Day Workshop on Function Theory 6th-9th September 2010 University College London Rod Halburd Flickr sets: Day 1 Day 2 Day 3 Day 4
Frontiers of Nevanlinna Theory I 28th-30th March 2011 University College London Rod Halburd I presented a poster, Pairs of non-homogeneous linear differential polynomials at this meeting.
Flickr sets: Day 1 Day 2
Complex Analysis & Geometry Meeting 24th May 2011 The Open University Dan Nicks, Ian Short Flickr set
One Day Function Theory Meeting 5th September 2011 London Mathematical Society Ian Short I gave a presentation, Integer points of meromorphic functions at this meeting.
Flickr set

 

 

Useful links
Title Description
Jim Langley's research page Professor Jim Langley's personal homepage.
Papers published by Jim Langley Complete list of Jim Langley's publications, including PDFs of most of them.
Postgraduate notes on Complex Analysis (PDF) Jim Langley's postgraduate notes on complex analysis.
School of Mathematical Sciences University of Nottingham School of Mathematical Sciences homepage.
MathSciNet Searchable database of papers.

 

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