Critical symplectic geometry, Lagrangian cobordisms, and stable homotopy theory

辛几何起源于经典牛顿力学的一种几何方法，统一了以前不相连的思想，并在不可能有显式解的情况下提供了对动力系统的定性理解。更新的发现表明，辛几何在许多其他数学领域中发挥着关键作用，如代数几何、低维拓扑和表示论。这个项目将把同伦理论和范畴理论的思想融入辛几何，以证明关于辛流形和辛流形之间的映射的结构结果。该项目还具有重要的教育内容。PI将在波士顿大学组织研究生院小组和数学推广活动，监督本科生研究，指导其他大学的研究生和博士后，并担任全国数学竞赛的评委。PI将研究临界辛几何：研究某些被称为温斯坦域直到稳定化和亚临界句柄的辛流形。PI在前人的工作中引入了临界辛几何，以定义有理同伦理论中拓扑局部化的辛类比，并推广了辛柔化。此外，临界辛几何是研究J-全纯曲线不变量的自然背景，比如Fukaya范畴，它在这两种运算下是不变的。在这个项目中，PI将把临界辛几何与拉格朗日余边线联系起来，展示拉格朗日余边线可以检测辛柔性，发展一种几何方法来研究Floer同伦理论，并研究来自Anosov动力系统的非Weinstein例子。该项目将使用同伦理论、高等代数和动力系统的现代技术，并将辛几何的思想引入这些数学领域。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Symplectic geometry originated as a geometric approach to classical Newtonian mechanics, unifying previously disconnected ideas and providing a qualitative understanding of dynamical systems in cases where explicit solutions are not possible. More recent discoveries have revealed that symplectic geometry plays a key role in many other fields of mathematics like algebraic geometry, low-dimensional topology, and representation theory. This project will incorporate ideas from homotopy theory and category theory into symplectic geometry in order to prove structural results about symplectic manifolds and maps between them. The project also has a significant educational component. The PI will organize graduate school panels and math outreach events at UMass Boston, supervise undergrad research, mentor graduate students and postdocs are other universities, and serve as a judge for nationwide math competitions.The PI will investigate critical symplectic geometry: the study of certain symplectic manifolds called Weinstein domains up to stabilization and subcritical handles. Critical symplectic geometry was introduced in the PI's previous work in order to define a symplectic analog of topology localization in rational homotopy theory and generalize symplectic flexibilization. Furthermore, critical symplectic geometry is the natural setting to study J-holomorphic curve invariants like the Fukaya category, which is invariant under these two operations. In this project, the PI will relate critical symplectic geometry to Lagrangian cobordisms, show that Lagrangian cobordisms can detect symplectic flexibility, develop a geometric approach to Floer homotopy theory, and investigate non-Weinstein examples arising from Anosov dynamical systems. The project will use modern techniques from homotopy theory, higher algebra, and dynamical systems and import ideas from symplectic geometry into these areas of mathematics.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.