Lagrangian submanifolds have classically played an important role in symplectic geometry. Recently their role has been expanded beyond that of their use in understanding symplectic diffeomorphisms. For example, conjecturally special lagrangian submanifolds play a central role in understanding the structure of Calabi-Yau 3-folds and in mirror symmetry (Strominger-Yau-Zaslow). Also, conjecturally they play a central role in understanding the relation between certain Gromov-Witten type invariants and knot invariants (Gopakumar-Vafa).
Although there has not been definitive progress on the construction of special lagrangian fibrations, the topological picture is much better understood (Gross, W.D. Ruan). From the point of view of existence theory for special lagrangian cycles, it has been shown that one can formulate a variational problem for volume and develop much of the necessary machinery (Schoen-Wolfson). In two dimensions the critical points of this variational problem can also be obtained, in some interesting cases, by the methods of completely integrable systems (Helein-Romon, Joyce). Recently geometric flow techniques, in particular, mean curvature flow, have been applied to lagrangian submanifolds (Thomas-Yau, M-T Wang) to construct special lagrangians in special cases. These lead to general conjectures about the nature of solutions of lagrangian mean curvature flow.
This workshop aims at exploring new applications of lagrangians in symplectic and Kähler geometry, with particular emphasis on techniques for construction.
(University of California at San Diego)
Kefeng Liu (UCLA)
Rick Schoen (Stanford University)
Jon Wolfson (Michigan State University)
Eric Zaslow (Northwestern University)