Introduction

Click on the tabs below for a (hopfully) accessible introduction to this area of research.

• Special Functions
• Abelian Functions
• Applications

Special functions and elliptic functions

Special functions can be loosely defined as those mathematical functions with established names and notations due to their importance. This is often apparent from the formulae and properties they satisfy and their use in other branches of mathematics. A simple example would be the trigonometric functions.

The high point of special function theory in the later 1800s was the theory of elliptic functions. These are essentially those functions of a complex variable that have two independent periods, (the values it takes repeat as you move in one of two directions). Hans Lundmark's Complex Analysis webpage gives a nice introduction with some excellent visualisations making the key properties clear. For example, the plot here is an elliptic function with the values taken (and hence patterns) repeating both horizontally and vertically.

The Weierstrass ℘-function is a particular type of elliptic function with a simple form and many useful properties. My work on Abelian functions could be summarised as the higher genus generalisation of Weierstrass elliptic function theory.

Abelian functions associated with algebraic curves

An Abelian function is a meromorphic function that has multiple periods. The functions can be defined using the periodicity properties of algebraic curves, leading to the curves as a classification system.

Kleinian ℘-functions are particular Abelian functions defined in analogy to the Weierstrass ℘-function, and satisfying similar properties. The main difference is that they are now functions of g variables where g is the genus of the algebraic curve used.

I have worked extensively with such functions investigating the addition formulae, differential equations, power series expansions and other properties that they satisfy. The theory lies within the fields of complex analysis and algebraic geometry. Their study has involved both new theory and intelligent use of symbolic computation. They have applications throughout mathematical physics and are perhaps best know as solutions to the KP-Heirarchy of integrable systems .

Application of Abelian functions

Applications of Abelian functions theory include use in soliton theory, describing the Kummer surface, modelling the motion of a double pendulum and describing geodesics in black hole spacetimes.

My Research

The main topics of my research in this area are summarised below. The references refer to papers listed under Publications.

• Tetragonal curves

  A cyclic tetragonal curve of genus six

  As discussed in the introduction, Abelian functions are usually classified by the curves defining their periodicity properties. I started my PhD by extending the theory to a tetragonal curve, f(x,y)=0, specifically a cyclic tetragonal curve of genus six:

  f(x,y) = y⁴ - ( x⁵ + λ₄x⁴ + λ₃x³ + λ₂x² + λ₁x + λ₀ )

  A genus six surface is one with six holes. Pictures of models of such surfaces can be seen here and here .

  

  We have constructed sets of Abelian functions upon the Jacobian of and derived a number of interesting results, summarised in [EE09] This article was included in IOP Select - a selection of free articles from IOP Journals chosen for their novelty, significance and potential impact on future research.

  Extending the theory required both new theoretical results and intellignet use of symbolic computation. These new ideas were applied again to the study of higher genus trigonal curves as described in [England10]

• Benney reductions

  A tetragonal reduction of the Benney equations

  I collaborated with Dr. John Gibbons (Imperial College London) on constructions reductions of the Benney moment equations. Certain reductions may be parametrized by conformal maps from the upper half plane to a slit domain cut along arcs. The mappings are expressed as integrals of Abelian differentials and can be evaluated using derivatives of the Kleinian sigma-function. We applied the results discussed above to construct a reduction using the functions associated with the genus six cyclic tetragonal curve. These results led to the joint paper [EG09].

  

• Vector space bases

  Defining new Abelian functions with prescribed poles structures

  I considered the problem of identifying bases for the vector spaces of Abelian functions with given poles structures. The key problem is identifying enough suitable functions. One approach is by taking algebraic combinations of ℘-functions, a method employed in [EEO11] and examined in detail by [England11].

  While this approach is successful in identifying bases for specific curves, it does not generalise. This led me to consider an alternative approach to the basis problem, where new classes of Abelian functions are defined through the repeated, potentially infinite, application of operators. Motivated by the definition of the Weierstrass ℘-function using the Hirota operator, we defined generalised operators and new classes of Abelian functions in an analagous way. The details were given in [EA12a].

  

• Addition formulae

  Constructing addition formulae for Abelian functions

  One of the key properties we derive for Abelian functions is the addition formulae they satisfy. The identification of the bases offers one route to these. In the elliptic case (genus one) these addition formula describe the famous group addition law for points on an elliptic curve, (how two points on the curve may always be combined to find a third). This is the basis for elliptic curve cryptography.

  

  [EEO11] studied the two canonical genus three cases using bases along with sophisticated symbolic computation techniques to identify new addition formulae.

  A recent project inspired by these results went on to discover new addition formulae for the Weierstrass elliptic function itself. These results have been presented in [EEO14]

• Equivariant theory

  The equivariant theory of Abelian functions

  Another recent area of study has been the equivariant theory of Abelian functions. This is an alternative approach to the functions where the underlying curves are redefined so that both the curve and functions transform under a group action allowing for the use of representation theory. The approach reduces computational requirements. For example, when deriving sets of differential relations we find they break down into representations which can each be generated by only one equation. The single matrix equation below can represent the 55 individual equations between the functions stored in this text file.

  The equivariant theory was developed by Chris Athorne at the University of Glasgow, who I have collaborated with on extending the approach. We have already had success in mapping between the existing theories for the hyperelliptic cases and rewriting knows sequences of bases in an equivariant form, as detailed in [EA12b].

  

Useful Links

• • The theory of elliptic functions is well documented and there are many resources available on the web. I reccommend the introduction on Hans Lundmark's Complex Analysis webpage as it has some excellent visualisations making the key properties clear.
• • Abelian functions are less well known. The 1897 and 1907 texts by Henry Frederick Baker describe the origins of the theory. Both have been recently repreinted with the original versions out of copyright and available freely online here and here.
• • The modern approach to Abelian functions was pioneered by Buchstaber, Enolskii and Leykin in their 1997 paper, which has recently been updated and made available freely here.
• • This webpage hosts some technical results for Abelian functions associated to trigonal and tetragonal curves. There is the possibility of more being made available in a more structured way later.
• • This webpage contains details and proceedings from a 2010 conference on the theory and acts as a good survey of current research in the area.
• • More generally, SIGMA has a regularly updated list of conferences in the area of symmetry and integrability; confereces at which this research is often presented.
• • As discussed, much of this research involved heavy use of the computer algebra package Maple. Many of the calulations were performed in parallel on a cluster of linux machines using the Distributed Maple package. Wolfgang Schreiner's Distributed Maple website lets you download the program along with instructions and examples.