Investigations in the application of homotopy theory

这个项目涉及同伦理论中的三个独立问题。第一部分是对代数K-场论和模同伦理论的研究。我们将利用导出的完备化结构继续研究代数K-场论的表示理论模型，并尝试利用Motivic同伦理论和代数K-理论来研究有限复上的自由有限群作用上的Halperin-Carlsson猜想。第二种是同伦理论的内在，它将对更高的拓扑循环同调结构进行分析，以期得到更高色度的现象的良好几何版本。第三个问题是对“广义持久性”的各个方面的研究，它为研究有限度量空间提供了一种方法，因此从科学和工程中产生了许多数据集。具体地说，我们希望研究某些抖动的条形码不变量，这将有助于评估拓扑信号的“强度”。这个项目涉及某些几何技术的应用，称为拓扑学，应用于数学的其他部分，最终甚至超出数学范畴。拓扑技术可以系统地捕捉几何对象的定性特征，如环、球体和其他表面的存在，这使得它可以广泛地用作几何模式识别的工具。以这种方式解释，这些技术在各种领域都有应用，从数论到代数方程的求解，一端是数据分析中的问题。此外，我们将在同伦理论的研究中追求一个有趣的方向，我们希望在那里建立具体的和几何的特定结构，这些结构目前没有那么明确的定义。

This project deals with three separate problems in homotopy theory. The first is the study of algebraic K-theory of fields and motivic homotopy theory. We will continue the study of the representation theoretic model for the algebraic K-theory of fields using derived completion constructions, and attempt to use motivic homotopy theory and algebraic K-theory to study the Halperin-Carlsson conjectures on free finite group actions on finite complexes. The second is internal to homotopy theory, and will carry out an analysis of higher topological cyclic homology constructions in the hopes of obtaining good geometric versions of phenomena of higher chromatic levels. The third problem is the study of various aspects of "generalized persistence", which provides a method for studying finite metric spaces, and therefore many data sets coming out of science and engineering. Specifically, we wish to study barcode invariants of certain quivers which will help to assess the "strength" of a topological signal. This project deals with applications of certain geometric techniques, referred to as topological, to other parts of mathematics and eventually even outside of mathematics. Topological techniques capture qualitative features of geometric objects, such as the presence of loops, spheres, and other surfaces within them in a systematic way which allows it to be used as a tool for geometric pattern recognition in a broad sense. Interpreted this way, these techniques have applications in areas as diverse as number theory and the solution of algebraic equations on the one end to problems in data analysis on the other end. In addition, we will pursue an interesting direction in the study of homotopy theory, where we hope to make concrete and geometrical certain constructions which at the moment are less sharply defined.