Formalized mathematics is experiencing rapid growth, with mathematicians increasingly using interactive theorem provers like Lean to validate proofs. The Lean ecosystem, including the extensive mathlib library, has enabled collaborative formalization of over 210,000 proofs. However, the process remains labor-intensive, prompting the development of AI-based tools to automate and enhance proof validation. The dream is for AI to eventually complete intricate proofs beyond human capability. This PhD project, based at The University of Manchester's Department of Computer Science, addresses two fundamental research questions: (1) How well can current AI models really prove? (2) How can we guide AI models to improve their proving capabilities? The project will invent methods inspired by automated program testing to empirically assess proving hardness and develop fine-tuning and training procedures to enhance AI proof capabilities. Applicants should hold, or expect to achieve, at least a 2.1 honours degree or a master’s (or international equivalent) in a relevant science or engineering discipline. The position is fully funded for 3.5 years by the School of Engineering, with excellent candidates nominated for additional competence-based competition funding. The start date is October 2026, and flexible study arrangements (including part-time options) may be considered depending on the project and funding. To apply, candidates should contact the supervisors to discuss their academic background and motivation for the project. Applications must be submitted online, specifying the project title, supervisor name, funding status, previous study details, and contact information for two referees. Required supporting documents include transcripts, CV, a supporting statement, and an English language certificate if applicable. Incomplete applications will not be considered. The University of Manchester is committed to equality, diversity, and inclusion, actively encouraging applicants from all backgrounds and career paths. The research environment is collaborative and interdisciplinary, with opportunities to engage in cutting-edge work at the intersection of artificial intelligence, software engineering, and pure mathematics. For further information, reference materials and application links are provided, including resources on Lean, mathlib, and recent advances in neural theorem proving. For application queries, contact the admissions team at [email protected].

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.

This PhD project at The University of Manchester focuses on advancing the field of neural theorem proving, with a particular emphasis on formalized mathematics using the Lean interactive theorem prover. Formalized mathematics is experiencing rapid growth, with mathematicians increasingly using tools like Lean to validate and collaborate on mathematical proofs. The mathlib library, containing over 210,000 Lean proofs, exemplifies the scale and collaborative potential of this approach. However, the process of formalizing proofs remains highly labor-intensive, prompting the development of AI-based tools to automate and assist in proof generation and validation. The project addresses two core research questions: (1) How well can current AI models actually prove mathematical theorems? (2) How can we guide and improve the proving capabilities of AI models? To answer these, the research will develop methods inspired by automated program testing and empirical analysis of proof difficulty, as well as invent fine-tuning and training procedures to enhance AI proof capabilities. The project is supervised by Prof U Sattler, Dr D Winterer, and Dr A Mukherjee in the Department of Computer Science. Applicants should have, or expect to achieve, at least a 2.1 honours degree or a master’s (or international equivalent) in a relevant science or engineering discipline. The position is a 3.5-year PhD, with excellent candidates nominated for competence-based competition funding. The start date is October 2026. The university values diversity and inclusion, encourages applicants from all backgrounds, and offers flexible study arrangements. Prospective students are strongly encouraged to contact the supervisors before applying, providing details of their academic background and motivation. Applications must be submitted online, including all required documents such as transcripts, CV, supporting statement, and referee contact details. For further information or questions about the application process, applicants can contact the admissions team at [email protected].

This PhD project at The University of Manchester focuses on the formalization and automated proof of Boolean function conjectures using large language models (LLMs) and the Lean interactive theorem prover. Formalized mathematics is rapidly advancing, with mathematicians increasingly using tools like Lean to validate proofs and facilitate collaboration. The mathlib library, for example, now contains over 210,000 Lean proofs. However, the area of Boolean function analysis—a field with deep mathematical roots and significant potential for formalization—has been relatively overlooked. This project aims to address this gap by (a) formalizing standard results in Boolean analysis and (b) developing techniques to fine-tune LLMs for generating formal proofs of known theorems in this domain. The research will explore the intersection of formal mathematics, machine learning, and circuit complexity, with the ambition of enabling LLMs to prove or refute conjectures about Boolean functions, potentially transforming approaches to longstanding problems in circuit complexity. The ideal candidate will be passionate about formal mathematics and the mathematics underlying LLMs, with top grades in advanced theoretical computer science or mathematics courses, and experience in Boolean function analysis or circuit complexity theory. A strong background in machine learning theory and/or mathematics is highly desirable. The position is a 3.5-year PhD, with excellent candidates nominated for competitive, competence-based funding. The start date is October 2026. Applicants should have at least a 2.1 honours degree or a master’s (or international equivalent) in a relevant science or engineering discipline. The application process requires an online submission with all supporting documents, including transcripts, CV, a supporting statement, and contact details for two referees. Prospective students are strongly encouraged to contact the supervisors before applying to discuss their background and motivation. The university values diversity and supports flexible study arrangements. For further details, see the project and application links provided.