Problems in Geometric, Algebraic and Quantitative Topology

拓扑学是数学中一个非常定性的部分；它不能区分一个圆球和一个椭球、一个盘子或一个碗。拓扑学家所做的测量通常是由代数结构来调节的，因为通常的长度或体积的概念在这种情况下是没有意义的。这一建议的主旨是提高我们对代数测量和几何之间的联系的理解，并研究定量测量在拓扑中的作用。在拓扑学内部(例如，为了更深入地理解对称性)和在拓扑学之外(例如，对于微分几何以及可能的数据科学)都有预期的应用。这个项目是对几何拓扑学中基本问题的一次尝试。该领域的经典范例是将几何问题归结为代数问题，要么通过同伦理论，要么通过K-理论或二次型理论。通常情况下，最成功的是某种形式的惠特尼引理。这个建议涉及(1)涉及对称性的问题，其中相关的惠特尼引理已知是错误的；(2)关于在代数处理中回避的拓扑问题的解的定量方面的理解。这些方法涉及到古德威利的同伦函子演算、控制拓扑学、分层运算、有理同伦理论和指数理论。除了它在拓扑学中的影响外，这项工作还可以扩大我们对函数空间几何的理解，包括分析中的含义，以及间接的数据分析的几何方法。

Topology is a very qualitative part of mathematics; it does not distinguish a round ball from an ellipsoid or a plate or bowl. The measurements done by topologists are usually mediated by algebraic structures, as the usual notions of length or volume are meaningless in this setting. The main thrusts of this proposal are to improve our understanding of the connection between the algebraic measurements and the geometry and to study the role quantitative measurements into topology. There are anticipated applications within topology (e.g., to a deeper understand of the symmetry) and outside of it (e.g., to differential geometry and possibly data science). This project is an attempt to deal with fundamental problems in geometric topology. The classical paradigm in that field is to reduce geometric problems to ones in algebra, either via homotopy theory or via K-theory or quadratic form theory. Typically involved, at its most successful, is some version of a Whitney lemma. This proposal deals with (1) problems involving symmetry, where the relevant Whitney lemma is known to be false and (2) with the understanding of quantitative aspects of solutions to topological problems that are sidestepped in the algebraic treatments. The methods involve Goodwillie's calculus of homotopy functors, controlled topology, stratified surgery, rational homotopy theory, and index theory. Besides its impact within topology, this work could expand our understanding of the geometry of function spaces, with implications in analysis, and also, indirectly, geometric methods of data analysis.