Geometry from Donaldson-Thomas invariants

长久以来，纯数学和理论物理之间有着密切的关系。数学的许多分支领域都是从试图解决理论物理中的问题开始的。一个著名的例子是牛顿对微积分的发展，他用微积分来理解行星的运动。另一方面，数学为物理学家描述他们的理论提供了一种基本语言，并为他们做出精确预测提供了计算工具。在过去的几十年里，数学和物理之间的这种关系变得极其深刻和重要。当今的相互作用围绕着一个被称为量子场论的主题，这是理论物理学中一个令人难以置信的强大计算工具，但它还没有被精确的数学术语所理解。量子场论被描述为无限维的微积分。本文讨论的是纯数学中的一类问题，这些问题与量子场论中的重要思想密切相关。我们的目的是了解这些问题在特殊情况下的解决办法，并证明一个表明它们总是可以解决的一般结果。与理论物理学家合作，并试图用数学术语重新表述他们的想法是这项工作的重要组成部分。我们希望这项研究不仅能带来新的、有趣的数学，还能带来量子场论的新见解。

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.