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，在这一领域的研究集中在量子不变量的研究定义使用几何对象称为全纯曲线。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。研究人员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，空间或时间的方向被重新标度。 应用将在拓扑和物理感兴趣。研究员亦会继续他在数学教育方面的辅导和推广活动，研究辛几何中量子不变量的三种绝热极限。 首先，他将研究极限的福谷类别的辛流形下的多方向辛场理论的限制，这是相当于在许多情况下收缩纤维的拉格朗日环面纤维化。 研究人员将以前的结果扩展到拉格朗日兼容与此热带极限，并将结果应用到镜像对称中产生的例子，如磁盘电位的计算。 其次，研究福谷范畴的行为和量子上同调下翻转与非平凡中心，或等价的，平均曲率流，其中一个子集的辛流形崩溃在一些非平凡的基础，目的是构造生成器的福谷范畴。 规范理论中的一个相关项目将研究高秩同调在λ参数变化下的行为，并将这些不变量与瞬子同调和阿贝尔同调联系起来。 在第三个项目中，研究人员将研究流空间的限制，这些流空间是由余切丛中的拉氏同构拉氏函数产生的，因为拉氏函数通过重新标度接近零截面，尤其是高-该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响进行评估来支持审查标准。

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.