Quantitative Topology and Embedding Theory

拓扑学领域拥有强大的工具来构造或阻碍几何空间和它们之间的连续映射的存在。 在这里，“连续”描述了一个空间到另一个空间的映射，可以被认为是弯曲和压缩，但没有断裂。例如，圆环面（甜甜圈的表面）可以被压缩，而不会以一种不会连续变形到一个恒定映射到一个点的方式撕裂到球体的表面上。然而，相反方向上的任何连续映射（将球体带到环面）都可能发生变形，而不会撕裂到单个点上的映射。拓扑学的经典工具没有解决这种结构中的大小或直径问题，本项目试图通过引入拓扑学和几何学的其他思想来研究这些问题。 这些项目的更广泛的影响包括指导资助的研究生。这些项目强调度量拓扑中的失真概念，并将带来微分分次代数，几何包装结构，配边和嵌入理论的技术。 其中的一些问题，包括嵌入，已经通过降维和持久同源性的研究进入了高维数据的研究。 拓扑学的代数学和几何学方面都涉及到这些调查，在定量几何拓扑学中出现的已知问题是逻辑意义上的复杂性和可判定性问题。该奖项反映了NSF的法定使命，并被认为是值得支持的，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。

The field of topology possesses powerful tools for constructing or obstructing the existence of geometric spaces and continuous maps between them. Here, "continuous" describes a mapping of one space to another that can be thought of as bending and compressing, but without breaking. For example, a torus (the surface of a doughnut) can be compressed without tearing onto the surface of a sphere in a manner that does not deform continuously to a constant map to a single point. However, any continuous map in the opposite direction, taking the sphere to the torus, may be deformed without tearing to a mapping onto a single point. Classic tools of topology do not address questions of size or diameter in such constructions, and this project seeks to study those questions by bringing other ideas of topology and geometry to bear. Broader impacts of these projects include mentoring of grant-supported graduate students.These projects emphasizes notions of distortion within metric topology, and will bring to bear techniques of differential graded algebras, geometric packing constructions, cobordism, and embedding theory. Some of these issues, including embedding, have entered the study of high-dimensional data through studies of dimension reduction and persistent homology. Algebraic and geometric aspects of topology are both involved in these investigations, and among the known issues arising in quantitative geometric topology are questions of complexity and decidability in the logical senses.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.