Geometry from Donaldson-Thomas invariants

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There has long been a close relationship between pure mathematics and theoretical physics. Many subfields of mathematics began with attempts to address problems from theoretical physics. A famous example is Newton's development of calculus, which he applied to understand the motion of the planets. On the other hand mathematics provides an essential language for physicists to describe their theories, and calculations tools for them to make precise predictions.In the last few decades this relationship between maths and physics has become extremely deep and important. The present-day interaction revolves around a subject called quantum field theory, which is an incredibly powerful calculational tool in theoretical physics, but which has not yet been understood in precise mathematical terms. Quantum field theory has been described as being the calculus of infinite dimensions.This proposal is about a class of problems in pure mathematics which are closely related to important ideas in quantum field theory. Our aim is to understand the solutions to these problems in particular cases, and to prove a general result which shows that they can always be solved. Collaborating with theoretical physicists, and trying to reformulate their ideas in mathematical terms is an important part of this work. As well as leading to new and interesting mathematics, our hope is that this research will lead to new insights in quantum field theory.