Symplectic Surfaces, Lefschetz Fibrations, and Arboreal Skeleta

University Of California-Davis, Davis CA

Investigators

Abstract

This award supports research in a rapidly growing field of mathematics known as symplectic topology that grew out of studying Hamilton's equations of motion in physics. Position and momentum coordinates are recorded in a higher dimensional space and the equations connecting position and momentum are kept track of in a geometric structure on the space. Powerful techniques were developed in the 1980s and 1990s to study these symplectic geometric spaces, but many fundamental questions remain unsolved. For example, we do not yet understand all of the surfaces that can be found inside these spaces. This project will explore such fundamental questions and develop new techniques for understanding symplectic spaces and for extracting their properties. The PI will share these new techniques with diverse groups of young mathematicians to bring the entering generation of students into cutting edge research. Activities included in this project will provide training in effective communication for students, to promote global collaborations, to engage the public in mathematics, and to ensure our scientific progress can effectively build off of work done by our predecessors and peers. There are three main scientific goals of this project. The first is to make progress on the longstanding symplectic isotopy problem, studying symplectic surfaces in the complex projective plane. The new approach proposed is to reduce to singular surfaces, which allow one to obtain large degrees without large genus and thus avoid the technical roadblock that stalled progress in the early 2000s. The second goal is to utilize Lefschetz fibrations to study questions arising in algebraic geometry and singularity theory from the symplectic topological perspective. Lefschetz fibrations have been an effective tool to construct and calculate invariants of symplectic manifolds, and the PI plans to apply this effective tool to open conjectures and new lines of research related to complex algebraic geometry. The third goal is to develop the emerging technique of arboreal skeleta to study Weinstein manifolds through topological methods utilizing a simple collection of singularity types. The PI has begun initial stages of this program in a recently published paper and will continue development of the theory and applications particularly in low dimensions. 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.