Higher structures in generalized geometry and mathematical physics

物理学和数学的进步是齐头并进的。 在量子场论中发现越来越复杂的物理系统，导致了新的和有趣的几何结构类型的发展。 几何学中的数学突破受到物理学建议的启发，或者出于同样的原因，它激发了新的物理思想，这并不罕见。 在这个意义上，每个主体都依赖于另一个主体。拟议研究的主要目标是在我们对广义几何的理解方面取得根本性进展，这是希钦在21世纪初引入的一种几何形式，在形式上类似于微分几何的伟大经典工具，如复几何，辛几何和黎曼几何，但与它们有显著的不同：几何形状出现的实际空间不是平滑的，而是“堆叠的”，这意味着它们在某种意义上是错综复杂的折叠和混乱的。 在2014年，我完成了广义Kahler几何的主要研究，这是一种新形式的广义几何，它来自于我早期与Hitchin建立广义复几何基础的工作。为了得到我的结果，我需要将联合收割机的方法结合起来，并在数学的几个领域引入新的思想，如泊松几何、狄拉克几何和厄米几何，从而引起几个不同领域和团体的兴趣。 更引人注目的是，我证明了广义Kahler几何是20世纪80年代物理学家引入的一类物理模型（称为二维sigma模型）所需的关键几何输入。 这导致了双方的快速发展，物理学家迅速利用我的工作中包含的新工具，数学家，包括我自己，证明了解决几十年来物理学问题的新结果。物理学家们预言的许多重要性质和结构还没有被证实，显然需要新的思想来解决我们面临的主要问题。 在这个建议中，我描述了我计划如何解决这些问题，以及我如何相信这项工作将反馈到物理学，为这两个领域的进步做出贡献。几何学和理论物理学之间的接口是一个快速而深刻的科学进步领域。 该提案涉及在这个界面上维持和发展整个研究小组，为加拿大学生和研究人员创造参与和在全球一级竞争的机会。

Progress in physics and mathematics go hand in hand. The discovery of increasingly complicated physical systems in quantum field theory has led to the development of new and interesting types of geometric structure. It is not unusual for a mathematical breakthrough in geometry to be inspired by a suggestion from physics, or by the same token, for it to inspire new physical ideas. In this sense each subject is dependent on the other. The main goal of the proposed research is to make fundamental advances in our understanding of generalized geometry, a form of geometry introduced by Hitchin in the early 2000s which is formally similar to the great classical tools of differential geometry such as complex, symplectic, and Riemannian geometry, but differs from them in a striking way: the actual space where the geometry occurs is not smooth, but rather "stacky", meaning that they are intricately folded and chaotic in a certain sense. In 2014, I completed a major study of generalized Kahler geometry, a new form of generalized geometry which emerged from my earlier work with Hitchin establishing the foundations of generalized complex geometry. To obtain my results, I needed to combine methods and introduce new ideas in several areas of mathematics, such as Poisson geometry, Dirac geometry, and Hermitian geometry, drawing interest from several different fields and communities. More strikingly, I showed that generalized Kahler geometry is the key geometric input needed for a class of physical models introduced by physicists in the 1980s, known as 2-dimensional sigma models. This led to a rapid development on both sides, with physicists quickly utilizing the new tools contained in my work, and mathematicians, myself included, proving new results which solved decades-old problems in physics.While we have made a great deal of progress, generalized Kahler geometry remains mysterious in several key ways; many important properties and conjectures predicted by physicists have not been established, and it is clear that new ideas are needed to solve the major problems which we face. In this proposal, I describe how I plan to attack these problems, and how I believe this work will feed back to physics, contributing to the progress of both fields. The interface between geometry and theoretical physics is an area of rapid and profound progress in science. This proposal involves the maintenance and growth of an entire research group at this interface, creating opportunities for Canadian students and researchers to participate and compete at a global level.