[School of Natural Sciences PhD Scholarships] Complex dynamics: Dynamics of analytic finite-type maps

About the project

About the Project A dynamical system is a system that changes over time. Such a system may be described by a set of possible states of the system, together with a transition rule, which determines how the system changes from one state to the next. In one-dimensional holomorphic dynamics, the state is described by a single complex number, and the transition rule by a holomorphic function. This may seem like an extremely simple setting, but many complex phenomena that occur in higher-dimensional systems can already be observed and analysed here. The goal of this project is to develop new results in the area of transcendental dynamics. Here the function describing the transition rule is not a polynomial or rational map, but has "essential singularities" (in a suitable sense) at the boundary of its domain of definition. Examples of such functions are functions such as exponential or trigonometric functions. The following describes a specific project, but this could be tailored to some degree to the prior experience of the successful applicant. A "finite-type map" between Riemann surfaces is one for which the inverse has only finitely many singularities. This is a very general class of functions, which includes rational self-maps of the Riemann sphere, the exponential and trigonometric functions mentioned above but, for example, also maps from the disc to the sphere such as elliptic modular functions. The dynamical theory of these maps was developed by Epstein in the 1990s, but interest in their dynamics has increased only recently. Nonetheless, there are many properties of these functions that remain unexplored. The goal of this project will be to develop a detailed understanding of two important parameter spaces: that of Klein's modular invariant (considered as a map from the unit disc to the complex plane) and that of certain parabolic renormalisations of quadratic polynomials (which play a crucial role in understanding the geometry of the famous Mandelbrot set). Formalisation of mathematics, using systems such as Lean, is becoming increasingly important in mathematics, as is the use of generative AI tools to help both with mathematical exploration and the formalisation of proofs. There has been comparatively little work within one-dimensional holomorphic dynamics in this direction, and the successful applicant will have the opportunity to participate in the formalisation of both classical and modern results in holomorphic dynamics. This project is expected to start in September 2027. Before you apply: We strongly recommend that you contact the supervisors for this project before you apply. How to apply: To be considered for this project you must complete a formal application through our online application portal. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships . If you don’t already have an applicant account, please follow the instructions here . When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications . Please note late or incomplete applications will not be considered. Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status. We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder. Eligibility : The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in Mathematics OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Mathematics. A strong background in complex analysis is necessary for this project, and prior background in holomorphic dynamics is desirable. This project will remain open until filled. If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026. If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027. Self or externally funded students can also be considered for this project. FSESoNS

Research areas