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 very 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 highly significant developments across mathematics before Wiles's celebrated proof was finally published in 1995. Wiles's proof 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. The Centre is needed to educate the next generation of academic researchers to maintain the excellence and competitiveness of the UK's universities and also to deliver highly trained mathematicians ready to take their place in financial and other high-tech industries. As shown by our letters of support from the Bank of England, the Satellite Applications Catapult, Heilbronn, Royal Bank of Scotland, and Schlumberger, a wide range of employers have the vision to invest in highly trained pure mathematicians. Our partners all speak highly of the analytical and problem-solving abilities of pure mathematicians trained to PhD level. The students trained in this Centre will be even more highly skilled: the structure of the training programme will encourage independence and leadership and will embed professional development and key skills such as programming, communication skills and public engagement alongside cutting-edge research in topics chosen from geometry and number theory.