Homotopy Algebras and Homotopy Theory

这个项目的目的是探索同伦代数的一些应用和潜在的应用。在代数K-理论中，该建议描述了在K-理论色塔上适用于Rognes猜想和Waldhausen猜想的思想，并将算术和几何联系起来。在不稳定同伦论中，该建议描述了E-无穷次微分分次代数及其相关同伦代数如何用于研究空间的同伦论。在稳定同伦理论中，讨论了E-n代数结构的阻塞理论，以及E-n代数结构与模范畴上结构的关系；同伦理论研究了数学对象在小变形下不变的性质。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。同伦的理论性质往往可以通过利用小变化下的不变性来进行计算。由于同伦理论通常也保留了关于原始数学对象的重要信息，因此同伦理论是解决广泛数学问题的有效工具。

The goal of this project is to explore some applications and potential applications of homotopy algebras. In algebraic K-theory, the proposal describes ideas applicable to the conjectures of Rognes and the conjecture of Waldhausen on the K-theory chromatic tower, relating arithmetic and geometry. In unstable homotopy theory, the proposal describes how E-infinity differential graded algebras and related homotopy algebras can be used to study the homotopy theory of spaces. In stable homotopy theory, the proposal discusses obstruction theory for E-n algebra structures, and the relationship of E-n algebra structures with structures on categories of modules.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.