Applications of Microlocal Sheaves of Spectra and K-Theory

本研究项目涉及代数拓扑学和辛几何。代数拓扑学是关于空间的那些因大的连续变形而不变的性质的理论；例如，有一个洞的性质。代数拓扑学家使用被称为环谱或特殊同调理论的数字系统来组织他们的学科。辛几何是经典力学的一种数学形式。辛几何中的许多问题都是用拉格朗日Floer理论来解决的。这个项目旨在开发一个单一的工具来概括这两个理论，在主题之间建立新的联系。Floer同伦理论是Floer同伦的一个新的精化。它与Floer同调具有相同的关系，就像特殊同调与普通同调具有相同的关系。归根结底，它应该把一个光谱丰富的范畴附加到辛流形上。这个项目的目的是利用微局域理论构造这个范畴，给出计算它的新技术，用它来证明关于辛流形的代数拓扑的一些猜想，并反过来辛地阐明这个范畴的某些方面。这个奖项反映了NSF的法定使命，并通过使用基金会的智力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.