Problems in Geometric, Algebraic and Quantitative Topology

拓扑学是数学中非常定性的一部分，它不能区分圆球和椭球、盘子或碗。 拓扑学家所做的测量通常是由代数结构介导的，因为通常的长度或体积的概念在这种情况下是毫无意义的。 这个建议的主要目的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学中的作用。在拓扑学中有预期的应用（例如，以更深入地理解对称性）和在其之外（例如，微分几何和可能的数据科学）。这个项目是试图处理几何拓扑学中的基本问题。该领域的经典范例是通过同伦理论或通过K理论或二次型理论将几何问题简化为代数问题。最成功的例子是惠特尼引理的某些版本。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是已知的错误和（2）与理解的定量方面的解决方案的拓扑问题，回避了代数处理。这些方法涉及Goodwillie的演算同伦函子，控制拓扑，分层手术，合理同伦理论和指数理论。 除了它在拓扑学中的影响，这项工作可以扩大我们对函数空间几何的理解，在分析中的影响，也间接地，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.