Applications of Microlocal Sheaves of Spectra and K-Theory

本研究项目涉及代数拓扑和辛几何。代数拓扑学是关于那些在大的连续变形下不改变的空间性质的理论；例如，有一个洞的性质。代数拓扑学家使用称为环谱或超同调理论的数字系统来组织他们的课题。辛几何是经典力学的一种数学形式。辛几何中的许多问题已经通过一种叫做拉格朗日花理论的工具得到了解决。这个项目旨在开发一个单一的工具来概括这两种理论，在学科之间建立新的联系。花同伦理论是花同伦的一个新兴的细化。它与花同源的关系就像异常同源与普通同源的关系一样。最终，它应该在辛流形上附加一个谱丰富的范畴。本课题旨在利用束的微局部理论构建谱的范畴，给出计算谱的新方法，并用它来证明关于辛流形代数拓扑的一些猜想，反过来对谱范畴的某些方面进行辛解释。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。

This research project concerns algebraic topology and symplectic geometry. Algebraic topology is the theory of those properties of space that are unchanged by large continuous deformations; for example, the property of having a hole. Algebraic topologists organize their subject using number systems called ring spectra or extraordinary homology theory. Symplectic geometry is a mathematical formalism for classical mechanics. Many questions in symplectic geometry have been resolved through use of a tool called Lagrangian Floer theory. This project aims to develop a single tool that generalizes both theories, forging new connections between the subjects.Floer homotopy theory is an emerging refinement of Floer homology. It bears the same relationship to Floer homology that extraordinary homology bears to ordinary homology. Ultimately it should attach a category, enriched in spectra, to a symplectic manifold. This project aims to construct this category using the microlocal theory of sheaves, to give new techniques for computing it, to use it to prove some conjectures about the algebraic topology of symplectic manifolds, and conversely to symplectically illuminate some aspects of the category of spectra.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.