[Stipend: ~£20,780/year (tax-free), tuition fees covered at UK rate, co-funded by DAS Ltd and the EPSRC Doctoral Landscape Award, research travel funds, optional paid part-time teaching assistantship.] This fully funded PhD opportunity at the University of Southampton focuses on the mathematics of deep bilevel learning, a cutting-edge area at the intersection of optimization, machine learning, and deep learning. Deep learning has revolutionized artificial intelligence, especially with the advent of large language models and transformer architectures. In parallel, bilevel optimization has become a powerful framework for modeling complex machine learning tasks, leading to the emerging field of deep bilevel learning. The project aims to investigate the mathematical foundations of deep bilevel learning, seeking to uncover structural properties that can be leveraged to design more efficient, robust, and explainable learning algorithms. The research outcomes have the potential to influence the next generation of AI systems and advance theoretical understanding in optimization and machine learning. The successful candidate will join a collaborative research environment, co-funded by DAS Ltd and the EPSRC Doctoral Landscape Award (Collaborative Studentships Scheme). The position offers a tax-free stipend of approximately £20,780 per year, with tuition fees covered at the UK rate. Additional benefits include research travel funds for conferences, possible visits to University College London and the Italian Institute of Technology, and the option for a paid part-time teaching assistantship. The supervisory team consists of Professor Alain Zemkoho (University of Southampton), an expert in bilevel optimization, and Professor Massimiliano Pontil (University College London & Italian Institute of Technology), a leading researcher in machine and deep learning. The project is ideal for candidates with a strong background in optimization, applied mathematics, and/or machine learning, who are motivated to shape the mathematical foundations of future AI technologies. Both UK and international applicants are encouraged to apply, though fees are covered at the UK rate. The PhD will commence on 1 October 2026 and has a duration of four years. The application deadline is 9 January 2026, but rolling interviews may be conducted until the position is filled. For further details and to apply, visit the University of Southampton's postgraduate research application portal for Mathematical Sciences. Enquiries can be directed to Professor Alain Zemkoho at [email protected].