EPSRC Centre for Doctoral Training in Geometry and Number Theory at the Interface: London School of Geometry and Number Theory

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.

In an EPSRC-commissioned report by Deloitte, the impact of mathematical sciences research (MSR) was estimated as contributing 10% of UK jobs and 16% of UK gross value added (approximately £208 billion). The total GVA contribution in the UK in 2010 of MSR was £556 billion, or over 40% of total GVA. MSR underpins almost every aspect of the knowledge economy, and that economy requires ever more sophisticated theoretical ideas for continuing growth and competitiveness. The Deloitte report recognises also that the time-lag between curiosity-driven blue-skies research in MSR and technological innovation is often very long (many decades, typically) but when they do appear their impact can be enormous. The recent Bond Review, facilitated by EPSRC, makes a strong case for the impact of mathematics to the UK: 'Approximately 50% of the growth in productivity in the UK in recent decades has derived from the development and uptake of innovative processes and technologies. Innovation across the entire economy is a fundamental driver of living standards for the UK and mathematics is arguably the single most pervasive and powerful of all drivers of innovation in the world today.'

This CDT, which comprises a partnership between Imperial College London, King's College London and University College London, will deliver a high-level training programme in pure mathematics, integrating transferable skills activities as a central and challenging part of the programme. The students graduating from our CDT will thus have undergone a universal training which will equip them to respond to the widest possible range of future theoretical challenges, whether from environmental consultancy, hedge-fund management, intelligence agencies and software development, biotech companies, artificial intelligence (as highlighted recently by the PM) and visualisation of large data (as in priority area 30: Towards a Data-driven Future). Pure mathematics PhD graduates are therefore in high demand in a diverse range of sectors, as evidenced by our list of industrial partners, and, by taking up such roles in industry, they contribute directly to the competitiveness of the UK economy and quality of life.

Given the extremely high quality of our current students and their significant academic achievements thus far, we expect a significant proportion are likely to find employment in academia: of the 14 graduating in the summer, 9 already have confirmed postdoctoral positions. This indicates the success of the CDT in terms of recruitment, training and development of students into potential future research leaders, which will undoubtedly contribute directly to the future educational and academic needs of the UK over the coming decades. We also have several students who have either confirmed or are considering taking up positions in industry, with Credit Suisse, Goldman Sachs and Morgan Stanley as the potential or chosen employers. In this way, we also see the direct contribution of the LSGNT to the financial sector, in particular.

We have direct links to GCHQ, through the Heilbronn Institute for Mathematical Research (HIMR), which supports an LSGNT PhD studentship each year, involving summer internships at HIMR, and which views geometry and number theory as of critical importance to their work. Through this strong support and fruitful interaction with HIMR, the LSGNT gives an invaluable contribution to national security and, more widely, to applied research.

The LSGNT students, as they graduate, represent future industrial partners who understand the value of pure mathematics, and with whom the CDT has a direct connection. The LSGNT can utilize this link, allowing a growing and potentially powerful dialogue to develop between pure mathematics and industry, with mutual benefits for both, which is strongly aligned to the recommendations of the Bond Review.