Award: DMS-0909273, DMS-0909021 Principal Investigator: Joshua M. Sabloff, Lisa Traynor
This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The principal investigators propose to investigate flexibility and rigidity questions that are central to the character of symplectic and contact topology by exploring knotting phenomena of Lagrangian and Legendrian submanifolds. Approaching symplectic and contact geometry through a topological lens has given rise to a young and thriving discipline with interesting questions that explore the boundary between flexibility (when the symplectic world behaves topologically) and rigidity (when the symplectic world behaves geometrically). In particular, the PIs plan to study the Arnold Conjecture for Legendrian submanifolds, squeezing phenomena for Lagrangian disks, and the cobordism theory of Lagrangian submanifolds. Special attention will be paid to Legendrian knots in contact 3-manifolds and Lagrangian surfaces in symplectic 4-manifolds. The methods used in addressing the aforementioned problems are of interest in their own right. On one hand, the PIs plan to develop and use pseudo-holomorphic techniques initiated by Gromov and Floer, and expanded to the Symplectic Field Theory (SFT) framework by Eliashberg, Givental, and Hofer. On the other, the PIs plan to develop and use generating family techniques. Investigations into the structure of the holomorphic and the generating family based invariants will yield insight into geometric flexibility and rigidity; making connections between these different types of invariants will increase understanding of each.
Symplectic and contact topology have their roots in physics as the language of classical mechanics and geometric optics. While remaining in touch with those roots, symplectic and contact topology have blossomed into a central mathematical field that combines features of geometry (the science of measurement) and topology (the study of the shape of space). This field has a variety of applications including fluid mechanics, differential equations, and the study of the possible shape of the 3-dimensional space and the 4-dimensional space-time in which we live. The goal of the PIs' project is to achieve a better understanding of how symplectic and contact topology sit between geometry and topology, and thereby strengthening the foundation for the aforementioned applications. The PIs'research activities will bring a diverse set of students at both the undergraduate and graduate levels into the process of discovering new mathematics, with the PIs' research with undergraduates serving as a pedagogical laboratory for integrating mathematical research into the PIs' curricula.
