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

几何和数论是纯数学的核心学科，在科学和社会中有许多影响。它们是自古希腊时代以来吸引了一些数学界最优秀人才的学科，并继续对专业和业余数学家产生自然的魅力。在整个数学史上，这两个主题往往激发了重大的数学发展，产生了巨大的影响，超出了原来的应用。数论的魅力可以从费马最后定理的故事中得到例证，费马最后定理的陈述写于1637年，它简单到任何熟悉高中数学的人都能理解。经过350多年的努力和数学领域的重大发展，怀尔斯的著名证明终于在1995年发表。怀尔斯的证明，他在2016年获得了著名的阿贝尔奖，涉及数论和几何的思想混合，这些主题之间的相互作用是当今纯数学研究中最活跃的领域之一。（为此他在2010年被授予菲尔兹奖，这是数学界的最高荣誉）和肖尔策在算术代数几何上的成就（为此，他在2016年获得了数学突破奖的新视野，并有望在今年获得领域奖章），显示了几何思想对数论的重大影响。在另一个方向，数论被用来证明几何中的猜想，包括Kontsevich（1998年菲尔兹奖，2015年突破奖）和Soibelman提出的一条路径，以帮助解决几何中的一个主要开放问题，SYZ猜想，它位于几何和理论物理的界面。几何和数论之间的这些和其他联系继续导致数学中一些最令人兴奋的研究发展。这个CDT将由伦敦帝国理工学院，伦敦国王学院和伦敦大学学院的研究人员合作运行，这些研究人员共同组成了英国最大和最强大的几何和数论中心之一。通过培养数学家到几何和数论博士水平，通过确保更多的一般技能（例如，计算，沟通，团队合作，领导力）嵌入作为我们的计划的要求和愉快的一部分，这CDT将提供下一代训练有素的研究人员，不仅能够为英国未来的教育需求做出贡献，而且也为金融和其他高科技行业。我们的毕业生将直接为国家安全做出贡献（例如，GCHQ是高端纯数学的用户），但也更间接地作为重视数学家通常带来的解决问题的创造性和新颖方法的行业的员工。

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 themost 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.