Loughborough University is a top-ten rated university in England for research intensity (REF2014) and an outstanding 66% of the work of Loughborough’s academic staff who were eligible to be submitted to the REF was judged as ‘world-leading’ or ‘internationally excellent’, compared to a national average figure of 43%. In choosing Loughborough for your research, you’ll work alongside academics who are leaders in their field. You will benefit from comprehensive support and guidance from our Doctoral College, including tailored careers advice, to help you succeed in your research and future career. In Mathematics and Physics the concept of symmetry is fundamental, often associated with the idea of invariance. The research on Automorphic Lie Algebras ultimately aims to describe symmetries. If you enjoy linear and abstract algebra, and if you appreciate the elegance of analysis and group theory, this project is for you. Symmetry has been a driving idea for art, architecture and music for centuries. Nature itself seems to be organised according to symmetries. In Mathematics and Physics the concept of symmetry plays a fundamental role, often associated with the idea of invariance (for example, under certain transformations). The research on Automorphic Lie Algebras (ALiAs) ultimately aims to describe symmetries. In fact, ALiAs could be described as continuous symmetries bearing a discrete symmetry themselves. More precisely, ALiAs are Lie algebras over a ring defined by invariance under the action of a finite group of automorphisms. They have been extensively studied in the last decade but only recently it was discovered that these algebras are intimately related with a cohomology theory on root systems. In this project, you will have the opportunity to work with world-leading experts in the theory of Automorphic Lie Algebras. In particular, you will adopt the latest theory to classify ALiAs associated to exceptional simple Lie algebras, starting from g_2, the smallest exceptional simple Lie algebra. If you enjoy linear and abstract algebra, and if you appreciate the elegance of analysis and group theory, this project is for you. Some ability to program and/or use computer algebra systems (such as GAP, Mathematica, etc.) might be an advantage.