Collaborative Research: RUI: Knotting Phenomena in Contact and Symplectic Topology

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AbstractAward: DMS-0909273, DMS-0909021Principal Investigator: Joshua M. Sabloff, Lisa TraynorThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The principal investigators propose to investigate flexibilityand rigidity questions that are central to the character ofsymplectic and contact topology by exploring knotting phenomenaof Lagrangian and Legendrian submanifolds. Approachingsymplectic and contact geometry through a topological lens hasgiven rise to a young and thriving discipline with interestingquestions that explore the boundary between flexibility (when thesymplectic world behaves topologically) and rigidity (when thesymplectic world behaves geometrically). In particular, the PIsplan to study the Arnold Conjecture for Legendrian submanifolds,squeezing phenomena for Lagrangian disks, and the cobordismtheory of Lagrangian submanifolds. Special attention will bepaid to Legendrian knots in contact 3-manifolds and Lagrangiansurfaces in symplectic 4-manifolds. The methods used inaddressing the aforementioned problems are of interest in theirown right. On one hand, the PIs plan to develop and usepseudo-holomorphic techniques initiated by Gromov and Floer, andexpanded to the Symplectic Field Theory (SFT) framework byEliashberg, Givental, and Hofer. On the other, the PIs plan todevelop and use generating family techniques. Investigationsinto the structure of the holomorphic and the generating familybased invariants will yield insight into geometric flexibilityand rigidity; making connections between these different types ofinvariants will increase understanding of each.Symplectic and contact topology have their roots in physics asthe language of classical mechanics and geometric optics. Whileremaining in touch with those roots, symplectic and contacttopology have blossomed into a central mathematical field thatcombines features of geometry (the science of measurement) andtopology (the study of the shape of space). This field has avariety of applications including fluid mechanics, differentialequations, and the study of the possible shape of the3-dimensional space and the 4-dimensional space-time in which welive. The goal of the PIs' project is to achieve a betterunderstanding of how symplectic and contact topology sit betweengeometry and topology, and thereby strengthening the foundationfor the aforementioned applications. The PIs'research activitieswill bring a diverse set of students at both the undergraduateand graduate levels into the process of discovering newmathematics, with the PIs' research with undergraduates servingas a pedagogical laboratory for integrating mathematical researchinto the PIs' curricula.