Quantitative Topology and Embedding Theory

定量拓扑和嵌入理论

• 批准号: 2105451
• 负责人: Shmuel Weinberger
• 金额: $ 36万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105451&HistoricalAwards=false
• 关键词: Quantitative; Topology; Embedding; Theory

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.

拓扑学拥有强大的工具来构造或阻碍几何空间及其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，但没有破裂。例如，圆环(圆环的曲面)可以在不撕裂到球体曲面的情况下以不连续变形到单个点的恒定贴图的方式进行压缩。但是，将球体带到圆环的相反方向上的任何连续贴图都可以变形，而不会撕裂到单个点上的贴图。经典的拓扑学工具并不解决此类建筑中的大小或直径问题，本项目试图通过引入其他拓扑学和几何学的思想来研究这些问题。这些项目的更广泛的影响包括对资助研究生的指导。这些项目强调度量拓扑中的扭曲概念，并将运用微分分次代数、几何填充构造、协边和嵌入理论的技术。其中一些问题，包括嵌入，已经通过降维和持久同调的研究进入了高维数据的研究。拓扑学的代数和几何方面都涉及到这些研究，在量化几何拓扑学中出现的已知问题中，有逻辑意义上的复杂性和可判断性问题。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。