OUPPS (Open University People Profile System)

Biography

Professional biography

I am currently the university's academic lead for Athena Swan, and Deputy Head of School with a particular interest in supporting the career development of staff.

Advancing women's careers in mathematics

I have a long standing interest in the issues surrounding women's careers in mathematics and chaired the London Mathematical Society's (LMS) Women in Mathematics Committee from 2006 to 2015. This work was recognized by the award of an OBE in 2015. Details of the work carried out by the LMS to support women's careers can be found here http://www.lms.ac.uk/women-mathematics. I was also a Diversity Champion on the EPSRC Mathematical Sciences SAT. I chaired the School's Athena SWAN self assessment team from 2013 to 2022 - we were awarded a Bronze award in 2014, which was renewed in 2017, and were awarded a Silver award in 2022.

Research interests

My research is in the area of complex dynamics and concerns the iteration of transcendental meromorphic functions. I am particularly interested in the possible dimensions of the Julia set, the structure of the escaping set and the behaviour of wandering domains.

Together with Professor Phil Rippon, I lead the complex dynamics group at the OU which is part of the rapidly growing dynamical systems group which has a regular seminar series.

The group includes Vasso Evdoridou (Lecturer) together with various PhD students and our activities often include other complex analysts in the department.

Phil Rippon and myself have received several grants from EPSRC, with the most recent funding a project to classify the behaviour in simply connected wandering domains. Together with Vasso, we continue to build on this work in collaboration with Nuria Fagella (University of Barcelona) and Anna Miriam Benini (University of Parma).

EP/R010560/1: Classifying Wandering Domains (ukri.org)

We were previously funded by EPSRC to investigate a surprising link that we identified between two open conjectures in complex dynamics: Baker's conjecture and Eremenko's conjecture. The work was funded by two grants which together funded our work for five years (0.5FTE each). More details on the project are given here:

EP/H006591/1: Baker's conjecture and Eremenko's conjecture: a unified approach
EP/K031163/1: Baker's conjecture and Eremenko's conjecture: new directions

Our former Research Associate, Dave Sixsmith, was funded by the following grant from the EPSRC:

EP/J022160/1: Dimensions in complex dynamics: spiders' webs and speed of escape

Teaching interests

Most of my teaching has been in the area of analysis. I am currently deputy chair for M208 Pure Mathematics and am responsible for the MSc course M835 Fractal Geometry.

Projects

CoA: Classifying wandering domains

Classifying Wandering Domains

The main aim of the proposed research is to complete the classification of the types of dynamical behaviour that can occur within the components of the Fatou set (the set of stability) arising in the iteration of analytic functions. There is a large historical body of work classifying the dynamical behaviour within periodic Fatou components and a recent paper gives a complete description of the behaviour in multiply connected wandering domains. This project will address the outstanding case of simply connected wandering domains. There are many different types of dynamical behaviour within such domains and they have not previously been studied systematically. The development of a unifying theory will be fundamental to future work on iteration. The theory will be illustrated and informed by the construction of new examples of wandering domains.

Baker's conjecture and Eremenko's conjecture: a unified approach (XM-08-066-GS)

Complex dynamics studies the iteration of analytic functions, to understand the structure of the Fatou set F(f) (stable set) and the Julia set J(f) (chaotic set). The escaping set I(f) consists of the points that iterate to infinity. Eremenko showed that I(f) is closely related to J(f), and asked key questions about it. As a result many authors have studied I(f), with major breakthroughs in recent years, many by the applicants. In 2007/8, we found key new techniques that should give much deeper understanding of the intricate structure of I(f). This application is to fund our time to achieve this.

Bakers Conjecture and Eremenko's Conjecture: New Directions (XM-12-066-GS)

This project is in the area of complex dynamics which concerns the repeated application or iteration of a function of the complex plane. This project concerns two major conjectures in the area, one due to Baker and one due to Eremenko. The investigators are currently funded by the EPSRC (2009 – 2012) to investigate a surprising connection that they discovered between these two conjectures. This work has led them to make two novel and significant hypotheses related to these conjectures. This proposal is for two additional years funding to investigate these new hypotheses and enable them to remain at the forefront of world leading research in complex dynamics.

Dimensions in complex dynamics: spiders' webs and speed of escape. (XM-11-079-GS)

This project is in the area of complex dynamics which concerns the repeated application or iteration of a function of the complex plane. One of the key objects of study in the area is the Julia set which consists of those points for which this iterative behaviour is chaotic. A closely related set is the escaping set which consists of those points that escape to infinity under iteration. It is of much interest to estimate the size of these sets, in particular, in terms of their dimensions. Most of the work to date in this area has focused on functions in a class B. This project will be groundbreaking in focusing on functions outside this class, particularly on functions for which the escaping set is known to have the novel structure of a spider’s web. The project will also investigate the size of sets of points that escape to infinity at different rates.