This fully funded PhD position at the University of Southampton offers an exciting opportunity to advance the mathematical foundations of deep bilevel learning—a cutting-edge area at the intersection of optimization and artificial intelligence. The project is co-supervised by Professor Alain Zemkoho (bilevel optimization expert, University of Southampton) and Professor Massimiliano Pontil (machine/deep learning expert, University College London & Italian Institute of Technology), providing a rich collaborative environment and potential for research visits to UCL and IIT. Deep learning has revolutionized AI, especially with the advent of large language models and transformer architectures. Bilevel optimization, meanwhile, has become a powerful framework for modeling complex machine learning tasks, leading to the emergence of deep bilevel learning. This project aims to investigate the mathematical underpinnings of deep bilevel learning, uncovering structural properties that can be leveraged to design more efficient, robust, and explainable learning algorithms. The research outcomes have the potential to shape the next generation of AI systems and contribute to theoretical advances in optimization and deep learning. The position is funded by DAS Ltd and the EPSRC Doctoral Landscape Award (Collaborative Studentships Scheme), offering a generous tax-free stipend of approximately £20,780 per year, plus research travel funds for conferences and optional paid part-time teaching assistantship. Both UK and international applicants are welcome, with tuition fees covered at the UK rate. Ideal candidates will have a strong background in optimization, applied mathematics, and/or machine learning, and be motivated to contribute to the mathematical foundations of future AI technologies. Applicants should prepare a personal statement detailing relevant experience (optimization, operational research, game theory, programming, mathematical background, and any industry collaboration), a CV, names and institutional email addresses of two academic referees, and all degree transcripts to date. Applications must be submitted online via the University of Southampton postgraduate research portal. Select 'Research' programme type, 2026/27 entry, Faculty of Social Sciences, and 'PhD Mathematical Sciences (Full time)'. In Section 2 of the application form, insert the name of the supervisor. The deadline for applications is Friday, January 9, 2026. This is a unique opportunity to work at the forefront of mathematical AI research, with strong academic mentorship and excellent funding support. For further details and to apply, visit the official project page or the university application portal.