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.