A consortium of mathematicians at Imperial College London, King’s College London, and University College London will train PhD students at the intersection of geometry and number theory—two of the oldest and most interconnected branches of pure mathematics. This matters because the interplay between geometry and number theory has driven some of the most profound advances in modern mathematics, from the proof of Fermat’s last theorem to breakthroughs in the Langlands program and arithmetic algebraic geometry. Yet the field requires sustained, specialist training to produce researchers who can push these connections further. The Centre for Doctoral Training addresses that gap by combining deep mathematical training with practical skills in computing, communication, and teamwork. The research is fundamentally curiosity-driven. It does not aim at an immediate practical application. However, past work in these fields has underpinned technologies such as cryptography, which secures online communications and financial transactions. Graduates of this programme will contribute to national security—GCHQ is a direct user of advanced pure mathematics—and to high-tech industries that value the creative problem-solving mathematicians bring. Deeper understanding of geometry and number theory could, over decades, open unexpected doors in areas like data security, quantum computing, or theoretical physics.
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Geometry and number theory are core disciplines within pure mathematics, with many repercussions across science and society. They are subjects that have attracted some of the best minds in mathematics since the time of the Ancient Greeks and continue to exert a natural fascination on professional and amateur mathematicians alike. Throughout the history of mathematics, both topics have often inspired major mathematical developments which have had enormous impact beyond their original applications. The fascination of number theory is exemplified by the story of Fermat's last theorem, the statement of which was written down in 1637 and which is simple enough to be understood by anyone familiar with high school mathematics. It took more than 350 years of hard work and significant developments across mathematics before Wiles's celebrated proof was finally published in 1995. Wiles's proof, for which he was awarded the prestigious Abel Prize in 2016, involves a mixture of ideas from number theory and geometry, and the interplay between these topics is one of the most active areas of research in pure mathematics today. For example, the work of Ngo on the Langland's program (for which he was awarded the Fields Medal in 2010, the highest honour in mathematics) and Scholze on arithmetic algebraic geometry (for which he was offered a New Horizons in Mathematics Breakthrough Prize in 2016, and is expected to be awarded the Field Medals this year), show the significant impact of geometric ideas on number theory. In the other direction, number theory has been used to prove conjectures in geometry, including a path proposed by Kontsevich (Fields Medal 1998, Breakthrough Prize 2015) and Soibelman to help solve one of the major open problems in geometry, the SYZ conjecture, which lies at the interface of geometry and theoretical physics. These and other connections between geometry and number theory continue to lead to some of the most exciting research developments in mathematics. This CDT will be run by a partnership of researchers at Imperial College London, King's College London, and University College London, which together form the largest and one of the strongest UK centres for geometry and number theory. By training mathematicians to PhD level in geometry and number theory, and by ensuring that more general skills (for example, computing, communication, teamwork, leadership) are embedded as a demanding and enjoyable part of our programme, this CDT will deliver the next generation of highly trained researchers able to contribute not only to the UK's future educational needs but also to those of the financial and other high-tech industries. Our graduates will contribute directly to national security (GCHQ is, for example, a user of high-end pure mathematics) but also more indirectly as employees in industries which value the creative and novel approach that mathematicians typically bring to problem solving.
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