Higher structures in generalized geometry and mathematical physics

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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.