Algebraic Topology and Algebraic K-Theory

本课题的目的是探索代数拓扑和代数k理论的一些应用和潜在应用。在代数k理论中，本项目探讨了适用于Waldhausen关于k理论色塔猜想的思想，涉及算术和几何。该项目探索了拓扑Hochschild同调和拓扑循环同调的理论和计算方面的一些潜在贡献。本项目探索了一些新的稳定同伦工具，这些工具适用于研究空间的代数k理论（与高维流形的微分拓扑密切相关）。在不稳定同伦理论中，本项目探讨了同伦代数在光纤平方和映射空间计算中的应用。同伦理论研究数学对象在小变形下不改变的性质。这些数学对象通常是几何性质的，但同伦理论的方法也越来越多地应用于代数性质的对象。利用小变化下的不变性，同伦理论性质易于计算。由于它们通常也保留了原始数学对象的重要信息，因此同伦理论是解决各种数学问题的有效工具。

The goal of this project is to explore some applications and potential applications of algebraic topology and algebraic K-theory. In algebraic K-theory, the project explores ideas applicable to the conjecture of Waldhausen on the K-theory chromatic tower, relating arithmetic and geometry. The project explores a number of potential contributions to the theory and computation of topological Hochschild homology and topological cyclic homology. The project explores some new stable homotopy tools applicable to the study of the algebraic K-theory of spaces (which is closely related to the differential topology of high dimensional manifolds). In unstable homotopy theory, the project explores the applications of homotopy algebras to computations related to fiber squares and mapping spaces.Homotopy theory studies those properties of mathematical objects that do not change under small deformations. These mathematical objects are often of a geometric nature but the methods of homotopy theory have been increasingly applied to objects of an algebraic nature as well. Homotopy theoretic properties tend to be accessible to computation by taking advantage of the invariance under small changes. Since they also generally retain important information about the original mathematical objects, homotopy theory is an effective tool for a wide range of mathematical problems.