CAREER: Aspects of Microlocal Geometry

在现代数学中，辛几何是经典力学的抽象背景，微局部分析是研究微分方程组解的工具。微局部轴理论是微局部分析的几何形式。最近的进展揭示了微局部轴理论在辛几何中的惊人应用。本项目将在各个方向探索和扩展这些应用，包括进一步发展曲线、结点和曲面之间的代数和组合连接。在教育方面，首席研究员将准备急需的关于微局部束理论的基本说明文。该提案的具体研究目标是双重的。首先，将温斯坦流形（某种精确辛流形）的深谷范畴的计算定位到它的拉格朗日骨架上。一旦定位，执行这个计算是一个组合问题；因此，第二，PI打算探索由此产生的组合学。我们已经知道，产生的组合学有表征论的味道；特别是出现了簇代数和分类结不变量。通过上述框架，探索它们与辛几何的联系，将获得这些主题的进一步见解。这些深谷范畴的定位和计算是本研究的重点，将在同调镜像对称猜想的方法中发挥重要作用。

In modern mathematics, symplectic geometry is the abstract setting for classical mechanics, and microlocal analysis is a tool used to study solutions to systems of differential equations. Microlocal sheaf theory is the geometric formulation of microlocal analysis. Recent advances have uncovered striking applications of microlocal sheaf theory to symplectic geometry. This project will explore and extend these applications in various directions, including further developing algebraic and combinatorial connections between curves, knots, and surfaces. In terms of education, the principal investigator will prepare much-needed elementary expository works on the subject of microlocal sheaf theory.The specific research goals of the proposal are twofold. First, to localize the calculations of the Fukaya category of a Weinstein manifold (a certain sort of exact symplectic manifold) to its Lagrangian skeleton. Once localized, performing this calculation is a matter of combinatorics; thus secondly, the PI intends to explore the resulting combinatorics. It is already known that the sort of combinatorics that arises has a representation-theoretic flavor; in particular cluster algebras and categorified knot invariants appear. Further insights in these subjects will be gained by exploring their connection with symplectic geometry in terms of the above-mentioned skeleta. The localization and calculation of these Fukaya categories that are the focus of this research will play an essential role in an approach to the homological mirror symmetry conjecture.