The Principal Investigator proposes to use singular Lagrangian fibrations to gain insight into the topology of symplectic manifolds. The ultimate goal is to develop an effective language for constructing and analyzing symplectic four-manifolds, analogous to Kirby calculus for smooth four-manifolds. This work lies at the intersection of toric geometry, integrable systems and smooth four-manifold topology. Progress on this project will allow the presentation of symplectic four-manifolds as generalized sums of well-understood manifolds. For instance, this approach has allowed the Principal Investigator to specialize a smooth surgery to the symplectic category and thereby determine the existence of symplectic structures on an infinite family of four-manifolds with exotic smooth structures. Further work will include extensions to dimension six where such fibrations arise in the ground-breaking theory of mirror symmetry.
A manifold is the generalization of a surface to other dimensions. For a manifold to be symplectic it must have an internal structure akin to the space of positions and velocities of a mechanical system such as a pendulum. Symplectic manifolds are ubiquitous in topology, geometry and physics. Recently great progress has been made in understanding symplectic manifolds, especially of dimension four, but many basic questions remain unanswered. For instance, given a manifold there is no general way to determine if it permits the internal structure required for it to be symplectic. One way to attack such a question is to require some additional structure on the manifold. The Principal Investigator proposes to deepen our understanding of these manifolds by appealing to a higher-dimensional analog of topographic maps for measuring elevation. The additional structure could be thought of as the analog of lines of constant elevation. In essence, the Principal Investigator plans to develop two-dimensional maps that yield enough information to completely determine the terrain (the four-dimensional manifold) and a legend that make these maps interpretable.
