EPSRC Centre for Doctoral Training in Algebra, Geometry and Quantum Fields (AGQ)

This CDT will create a cohesive, internationally-leading, cross-domain training and research community fusing algebraic, geometric and quantum methods across Algebra, Geometry and Topology, Mathematical Physics, and their Interfaces. The scientific aim of our CDT is no less than to develop new foundations unifying all three disciplines, and in the process to bolster and future-proof UK capability in mathematics. The breadth of mathematical mastery necessary to achieve these aims, on which our training programme is based, is of the highest international standard, and training students in this area requires both the deep focus and the wide scope which only the resources of a CDT can enable.

Our three scientific areas Algebra, Geometry and Quantum Fields are established, flagship, internationally-leading areas of UK mathematical strength. Algebra: quite simply *the* language, and controlling structure, of symbolic computation and symmetry. Geometry: the mathematically rigorous foundations of our human spatial and visual intuition. Quantum Fields: the mathematical incarnation of our quantum physical reality.

A hallmark feature of 21st century mathematics is the dramatically increased synergy and inter-dependence between these three fundamental disciplines. Whereas in centuries past mathematics and physics interacted primarily through analysis and calculus, the advent of quantum mechanics posits a fundamentally different, fundamentally algebraic, set of laws for the universe. Geometry enters irrevocably when we pose quantum mechanical laws in the presence of fields, such as the electro-magnetic and gravitational fields, which permeate throughout time and space. A surprising and thrilling discovery of 21st century mathematics has been that the mathematically rigorous study of quantum fields yields some of the most powerful predictive theories within algebra and geometry, even to questions with no a priori physical formulation.

These fundamental scientific developments have had a vast and direct impact on our modern world, and on a remarkably short timescale. Algebra, geometry and quantum fields are the driving force behind key developments such as internet search, quantum computation, machine learning, and both classical and quantum cryptography.

Society and industry need the students we will train. Our graduates' skills are both fundamentally transferable and widely applicable across many external partnerships and stakeholders. The Deloitte report, commissioned by EPSRC, attributed 2.8M jobs and £200BN of the UK economy to mathematical sciences research, highlighting R&D, computing/tech, public administration, defence, aerospace and pharmaceuticals as economic sectors requiring graduates with advanced mathematical training. Sustainable energy consulting has since emerged as a further industry requiring ever-advanced mathematical sophistication.

Crucially a physical and mathematical powerhouse needs to be a diverse powerhouse, yet the traditional structure of training in these areas has inhibited diversity of entrants, both to career academia and to industry. Building on our track record, and equipped with the resources and flexibility only a CDT can provide, we will create a diverse and confident cohort, equipped with the mathematical skillsets needed for our tech-led future to flourish, and able to influence a wide range of people, sectors and institutions.