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中的核心学科，在科学和社会中产生了许多影响。它们是自古希腊时代以来就吸引了一些最优秀的数学头脑的学科，并继续对专业和业余数学家产生自然的吸引力。纵观数学史，这两个主题往往激发了重大的数学发展，这些发展产生了巨大的影响，超出了它们最初的应用。数论的魅力以费马大定理的故事为例，它的陈述写于1637年，足够简单，任何熟悉高中数学的人都能理解。经过350多年的艰苦工作和数学上的重大发展，怀尔斯的著名证明终于在1995年出版。怀尔斯的证明涉及数论和几何的混合思想，他因此在2016年获得了著名的阿贝尔奖，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。例如，Ngo在朗格兰项目(他因此而在2010年被授予数学界最高荣誉菲尔兹奖)和斯科尔茨在算术代数几何(他因此而在2016年获得数学突破新视野奖，并有望在今年被授予菲尔德奖)的工作表明了几何思想对数论的重大影响。另一方面，数论已被用来证明几何猜想，包括康采维奇(菲尔兹奖1998年，2015年突破奖)和索伊贝尔曼提出的一条途径，以帮助解决几何中的一个主要开放问题，位于几何和理论物理的交界处的Syz猜想。几何和数论之间的这些和其他联系继续导致一些最令人兴奋的数学研究发展。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营，这三个中心共同构成了英国最大和最强大的几何和数论中心之一。通过培训数学家到几何和数论的博士水平，并确保更多的一般技能(例如，计算、沟通、团队合作、领导力)作为我们课程要求和享受的一部分，CDT将培养出训练有素的下一代研究人员，他们不仅能够为英国未来的教育需求做出贡献，而且还能够为金融和其他高科技行业的需求做出贡献。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)，但也更间接地成为行业的员工，这些行业重视数学家通常为解决问题带来的创造性和新奇方法。

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.