Quantitative Topology and Embedding Theory

拓扑学领域拥有构造或阻碍几何空间及其间连续映射存在的强大工具。这里，“连续”描述了一个空间到另一个空间的映射，可以认为是弯曲和压缩，但没有断裂。例如，一个环面（一个甜甜圈的表面）可以被压缩而不撕裂到一个球体的表面上，这种方式不会连续变形到一个单一点的恒定映射。然而，任何相反方向的连续映射，将球体映射到环面，可以在不撕裂的情况下变形为映射到单个点上。拓扑学的经典工具没有解决这种结构的尺寸或直径问题，本项目试图通过引入拓扑学和几何的其他思想来研究这些问题。这些项目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畸变概念，并将带来微分梯度代数、几何填充结构、共坐标和嵌入理论等技术。其中一些问题，包括嵌入，已经通过降维和持续同源的研究进入了高维数据的研究。这些研究都涉及到拓扑的代数和几何方面，在定量几何拓扑中出现的已知问题中，有逻辑意义上的复杂性和可决性问题。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。

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.