Adiabatic Limits of Quantum Symplectic Invariants

点击翻译按钮获取中文摘要

Symplectic geometry is the mathematical foundation for particle motion in classical physics. Recently, research in this area has centered on the study of quantum invariants defined using geometric objects known as holomorphic curves. These quantum invariants have appeared not only in fields of mathematics known as geometric analysis and low-dimensional topology but also in certain models in high-energy physics. The investigator will study the behavior of these quantum invariants under adiabatic limits in which directions in space or time are re-scaled. Applications will be of interest in topology and physics. The investigator will also continue his mentoring and outreach activities in mathematics education.The investigator will study three types of adiabatic limits of quantum invariants in symplectic geometry. First, he will study the limit of the Fukaya category of a symplectic manifold under the multi-directional symplectic field theory limit, which is equivalent in many cases to shrinking the fibers of a Lagrangian torus fibration. The investigator will extend previous results to Lagrangians compatible with this tropical limit and apply the results to examples arising in mirror symmetry, such as the computation of disk potentials. Secondly, the investigator will study the behavior of the Fukaya category and quantum cohomology under flips with non-trivial centers, or equivalently, mean curvature flow in which a subset of the symplectic manifold collapses over some non-trivial base, with the aim of constructing generators for the Fukaya category. A related project in gauge theory will study the behavior of higher rank monopole Floer homology under variation of monopole parameter, and relate these invariants with instanton homology and abelian monopole Floer homology. In a third project, the investigator will study the limits of flow spaces arising from Fukaya-isomorphic Lagrangians in cotangent bundles as the Lagrangians approach the zero section via rescaling, and in particular higher-dimensional moduli spaces of Morse flow trees that appear in the limit.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.