Principal Investigator: Christopher Woodward
The PI will carry out projects on functoriality for Lagrangian correspondences in Fukaya-Floer theory and functoriality for quotients in Gromov-Witten theory. The first group of projects will have applications in Gromov-Witten theory and ``cohomological'' mirror symmetry, that is, in the sense of Givental etc. With F. Ziltener and his former postdoctoral advisee E. Gonzalez he will investigate functoriality for Gromov-Witten invariants under the symplectic quotient construction. Potential applications include invariance of Gromov-Witten invariants under symplectic birational equivalence, to cohomological mirror symmetry for complete intersections of general type. With K. Wehrheim and his former student S. Mau the PI will study functoriality of Lagrangian correspondences in Floer-Fukaya theory. Applications include symplectic definitions of non-abelian Floer homology for tangles and arbitrary three-manifolds, possible generalizations of Khovanov homology and categorification of quantum groups. These projects will have applications in homological mirror symmetry and low-dimensional topology. Some of the projects have a graduate education component, and the PI also proposes several undergraduate research projects.
Overall the research carried out under this grant will advance the understanding of symplectic geometry, which is the mathematical language for classical dynamical systems, and the relationship between gauge theory, representation theory, and quantum physics. Gauge theories arise naturally in a number of physical settings, such as electromagnetism. The first part of the project concerns certain gauge theories with an addition "non-linear" field taking values in a classical phase space, which have been substantially studied in the physics literature in the linear case under the name "gauged sigma models". The second part of the project concerns the structural properties of "Floer-theoretical" invariants which have been extensively studied in relation to dynamical systems in recent years.
