Vortices, Quilts, and Quasimaps

The PI will use gauged Floer theory to study non-displaceability of Lagrangian tori in possibly open symplectic manifolds via compactifications such as symplectic toric orbifolds, and compare with results of McDuff on displaceability. With F. Ziltener, his former postdoctoral advisee E. Gonzalez, and his current student Sushmita Venugopalan the PI will study the quantum cohomology of quotients and relationships with gauged Gromov-Witten theory. In particular he will (a) prove a quantum version of Kalkman's wall-crossing formula which governs the behavior of Gromov-Witten invariants under variation of symplectic quotient and proves birational equivalence under crepant flops in many cases, as conjectured by Ruan (b) find presentations of quantum cohomology rings of symplectic quotients such as toric orbifolds and quiver varieties. With K. Wehrheim and his former student S. Ma'u the PI will continue to study holomorphic quilts, functoriality of Lagrangian correspondences in Floer-Fukaya theory, and relationship with low-dimensional topology. In particular, the PI will prove the A-infinity version and the exact triangle for fibered Dehn twists. These research projects will improve our understanding of symplectic geometry, which is a mathematical framework for understanding classical mechanics, particularly for time-dependent energy functions. Many of these projects are related to the behavior of symplectic invariants under the operation of symplectic reduction in which the number of degrees of freedom of a system is reduced by using symmetry. These invariants also appear in the study of quantum-mechanical ``sigma models'' by physicists, who have made a number or predictions about their behavior under symplectic reduction, some of which will be verified and generalized as part of the project. The project on holomorphic quilts will have applications to our understanding of three- and four-dimensional spaces, especially invariants constructed using gauge theories which are conjectured (and in some cases known) to have symplectic interpretations. The PI will also continue his involvement with middle school mathematics teachers.