G2-Instantons on Joyce-Karigiannis Manifolds

G2-instantons are solutions to a particular partial differential equation, the G2-instanton equation. The project aims to answer the question what the set of G2-instantons looks like.
The project focuses on one class of seven-dimensional spaces on which to consider the G2-instanton equation, called Joyce-Karigiannis manifolds. In particular, the questions to be answered are: Do there exist G2-instantons on these manifolds? If there exist any, are there finitely many or infinitely many?
One novel contribution is to transfer analytic results from unbounded spaces to Joyce-Karigiannis manifolds, in order to show existence of solutions. This process is called gluing, and has previously been carried out on other spaces. Another novel contribution is to use solutions to the well understood Hermitian Yang Mills equation in dimension six, to obtain a family of solutions to the G2-instanton equation. This part makes use of results from the field of algebraic geometry to obtain solutions in dimension six. It will also feed back into this field, by studying solutions in dimension six which have a particular symmetry.
There is an ongoing research effort in pure mathematics to count solutions to the G2-instanton equation in order to obtain a numerical invariant of a space. All of this is motivated by string theory, which predicts that there is some way to obtain a numerical invariant (called an observable) from counting solutions to the G2-instanton equation. This project has the potential not only to construct example G2-instantons, but to find all G2-instantons on a given space for the first time, owing to the special connection to six dimensions.
Furthermore, there is a conjecture about the limiting behaviour of families of solutions to the G2-instanton equation. This project may produce examples that can give further evidence to this conjecture.

In a recent EPSRC-commissioned report by Deloitte, the impact of mathematical sciences research (MSR) was estimated as contributing 10% of UK jobs and 16% of UK gross value added (approximately £208 billion). MSR underpins almost every aspect of the knowledge economy, and that economy requires ever more sophisticated theoretical ideas for continuing growth and competitiveness. The Deloitte report recognises also that the time-lag between curiosity-driven blue- skies research in MSR and technological innovation is often very long (many decades, typically) but when they do appear their impact can be enormous.

This CDT, which comprises a partnership between Imperial College, King's College London and University College London, will deliver a high-level training programme in pure mathematics, integrating transferable skills activities as a central and challenging part of the programme. The students graduating from our CDT will thus have undergone a universal training which will equip them to respond to the widest possible range of future theoretical challenges, whether from environmental consultancy, hedge-fund management, intelligence agencies and software development, biotech companies, artificial intelligence and visualisation of large data. We expect that approximately half of our graduates will take up such roles, and in so doing contribute directly to the competitiveness of the UK economy and quality of life. The other half are likely to find employment in academia, and thus will contribute directly to the future educational and training needs of the UK over the coming decades.