[School of Natural Sciences PhD Scholarships] Random walks and Fractals

About the project

About the Project This project will investigate random walks on groups and their applications to fractal geometry, with a particular focus on spectral gap phenomena and their implications for the absolute continuity and regularity of projections of self-similar measures. The project will build on the work of Lindenstrauss and Varjú on random walks on groups of Euclidean isometries and seek to develop these methods in the context of fractal measures. A central objective is to understand how spectral gap properties for the rotational component of a random walk can be established and subsequently exploited to obtain information about the associated self-similar measures. The project will study averaging operators associated with probability measures on groups, their spectral properties, and the way in which a spectral gap gives quantitative mixing and equidistribution. Particular emphasis will be placed on the interaction between the algebraic structure of the underlying group, Fourier-analytic estimates, and the geometry of self-similar constructions. The second major objective is to apply these techniques to projections of self-similar measures. Given an iterated function system of contracting similarities, the project will investigate conditions under which projections of the corresponding invariant measure are absolutely continuous, or possess stronger regularity properties. The approach will draw on the principle that sufficient expansion and mixing in the rotational part of the associated random walk can lead to decay of Fourier coefficients and hence regularity of the projected measure. A key aim will be to identify settings in which the Lindenstrauss–Varjú framework can be extended or refined, potentially yielding new absolute-continuity results. The student will receive training in the relevant areas of modern pure mathematics, including probability and random walks on groups, representation theory and harmonic analysis, geometric measure theory, fractal geometry, and Fourier analysis of measures. Initial work will involve a detailed study of the foundational literature, particularly the work of Lindenstrauss and Varjú, together with related developments in the theory of self-similar measures and their projections. Regular supervision meetings will be used to develop the necessary background, discuss research papers, formulate intermediate problems, and assess progress towards original results. As the project develops, the student will investigate specific classes of self-similar measures and groups for which spectral gap methods may be particularly effective. Potential outcomes include new spectral gap estimates, extensions of existing random-walk techniques, and new criteria for absolute continuity or quantitative regularity of projections of self-similar measures. The project is intended to develop into an original contribution to the interface between random walks on groups, harmonic analysis, and fractal geometry. 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. Previous research experience or a strong background in analysis and probability theory 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