Formalized mathematics is experiencing rapid growth, with mathematicians increasingly using interactive theorem provers such as Lean to formalize and validate proofs. This approach not only automates validation but also enhances collaboration, as demonstrated by the mathlib library, which contains over 210,000 Lean proofs developed in the past eight years. Despite these advances, Boolean function analysis—a concrete and fruitful area—has been largely overlooked in recent formalization efforts. Given the finite nature of Boolean functions and the depth of Fourier analysis on the Boolean hypercube, this project seeks to bridge this gap by formalizing standard results in Boolean analysis and developing techniques to fine-tune large language models (LLMs) for generating formal proofs of known theorems in this domain. This challenging PhD project aims to pioneer new methods for proving or refuting conjectures about Boolean functions using LLMs, potentially revolutionizing approaches to circuit complexity and mechanized mathematics. The research will combine elements of artificial intelligence, software engineering, and pure mathematics, with outcomes expected to include both new mathematical results and advances in formalized, mechanized mathematics. The ideal candidate will be passionate about formal mathematics and the mathematics underlying LLMs. Applicants must have top grades in advanced theoretical computer science or mathematics courses and should have completed coursework or possess research experience in Boolean function analysis or circuit complexity theory. A strong background in machine learning theory and/or mathematics is highly desirable. Eligibility requires at least a 2.1 honours degree or a master’s (or international equivalent) in a relevant science or engineering discipline. Non-native English speakers may need to provide an English language certificate. This is a fully funded 3.5-year PhD position supported by the School of Engineering at The University of Manchester. Outstanding candidates will be nominated for competence-based competitive funding. The anticipated start date is October 2026, and flexible study arrangements, including part-time options, may be available depending on project and funding requirements. Applications are accepted year-round. Prospective students are strongly encouraged to contact the supervisors prior to applying, providing details of their academic background, current level of study, relevant experience, and motivation for pursuing this PhD. Applications should be submitted online via the university’s portal, specifying the project title, supervisor, funding status, previous study details, and contact information for two referees. Required supporting documents include transcripts, CV, a supporting statement outlining motivation and relevant experience, and an English language certificate if applicable. For questions regarding the application process, contact the admissions team at [email protected]. The University of Manchester is committed to equality, diversity, and inclusion, welcoming applicants from all backgrounds and career paths, including those returning from career breaks. The research environment is designed to foster creativity, productivity, and societal impact through diversity.