LEAPS-MPS: The geometry of three-forms and their degenerations

Many fundamental laws of physics and nature are best described using differential forms. A notable example is the celebrated Maxwell's equations of electromagnetism, which is the foundation of almost all modern technologies. Differential forms carry a natural grading known as the degree. For degree 1 and degree 2 differential forms, their significance has been widely recognized as they are related to fundamental concepts like differentials, curvatures and symplectic structures. However, the importance of differential 3-forms has not received the attention it deserves. In this project, the PI proposes to systematically investigate the geometry associated to differential 3-forms. This research will find applications in understanding both complex geometry and symplectic geometries, as well as their degenerations. One key ingredient of this project involves training future researchers to address such problems. This will be accomplished by mentoring, teaching, developing new courses, as well as creating openly accessible documents describing cutting edge research results. This project will focus on the geometry of differential 3-forms in real six-dimensional manifolds. Recently, the principal investigator has discovered that there is a brand new type of geometric structures associated to certain unstable orbits of differential 3-forms. This viewpoint provides us a new framework in understanding degeneration of Calabi-Yau structures and it also leads to a novel perspective on the SYZ conjecture. Another approach the PI will take is to continue his study of the Type IIA flow of differential 3-forms with his collaborators. In particular, the PI wishes to establish certain compactness result for comprehending the singularities emerging from Type IIA superstring theories. As an application, the Type IIA flow can also be used to detect canonical geometric structures on symplectic 6-manifolds. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.