Vortices, Quilts, and Quasimaps

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The PI will use gauged Floer theory to study non-displaceability ofLagrangian tori in possibly open symplectic manifolds viacompactifications such as symplectic toric orbifolds, and compare withresults of McDuff on displaceability. With F. Ziltener, his formerpostdoctoral advisee E. Gonzalez, and his current student SushmitaVenugopalan the PI will study the quantum cohomology of quotients andrelationships with gauged Gromov-Witten theory. In particular he will(a) prove a quantum version of Kalkman's wall-crossing formula whichgoverns the behavior of Gromov-Witten invariants under variation ofsymplectic quotient and proves birational equivalence under crepantflops in many cases, as conjectured by Ruan (b) find presentations ofquantum cohomology rings of symplectic quotients such as toricorbifolds and quiver varieties. With K. Wehrheim and his formerstudent 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 PIwill prove the A-infinity version and the exact triangle for fiberedDehn twists.These research projects will improve our understanding of symplecticgeometry, which is a mathematical framework for understandingclassical mechanics, particularly for time-dependent energy functions.Many of these projects are related to the behavior ofsymplectic invariants under the operation of symplectic reduction inwhich the number of degrees of freedom of a system is reduced by usingsymmetry. These invariants also appear in the study ofquantum-mechanical ``sigma models'' by physicists, whohave made a number or predictions about their behavior undersymplectic reduction, some of which will be verified and generalizedas part of the project. The project on holomorphic quilts will haveapplications to our understanding of three- and four-dimensionalspaces, especially invariants constructed using gauge theories whichare conjectured (and in some cases known) to have symplecticinterpretations. The PI will also continue his involvement with middleschool mathematics teachers.