Adiabatic Limits of Quantum Symplectic Invariants

辛几何是经典物理学中粒子运动的数学基础。最近，这一领域的研究主要集中在研究使用全纯曲线的几何对象定义的量子不变量。这些量子不变量不仅出现在几何分析和低维拓扑等数学领域，而且出现在高能物理的某些模型中。研究者将研究这些量子不变量在绝热极限下的行为，在绝热极限下，空间或时间方向被重新缩放。应用将感兴趣的拓扑和物理。研究者还将继续他在数学教育方面的指导和推广活动。研究者将研究辛几何中量子不变量的三种绝热极限。首先，他将研究在多向辛场论极限下辛流形的Fukaya范畴的极限，这在许多情况下相当于收缩拉格朗日环面纤维。研究者将把之前的结果扩展到与热带极限相容的拉格朗日量，并将结果应用于镜像对称中出现的例子，如盘势的计算。其次，研究者将研究具有非平凡中心的翻转下的Fukaya范畴和量子上同调的行为，或者等效地，辛流形的子集在一些非平凡基上坍缩的平均曲率流，目的是构造Fukaya范畴的发生器。规范理论中的一个相关课题将研究单极子参数变化下高阶单极子Floer同调的行为，并将这些不变量与瞬子同调和阿贝珥单极子Floer同调联系起来。在第三个项目中，研究者将研究当拉格朗日量通过重新标度接近零截面时，共切束中fukaya同构拉格朗日量产生的流动空间的极限，特别是在极限中出现的莫尔斯流树的高维模空间。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.