Homotopy Algebras and Homotopy Theory

本课题的目的是探讨同伦代数的一些应用和潜在应用。在代数k理论中，本文描述了适用于Rognes猜想和Waldhausen关于k理论色塔猜想的思想，涉及算术和几何。在不稳定同伦理论中，本文描述了如何利用e -∞微分梯度代数和相关的同伦代数来研究空间的同伦理论。在稳定同伦理论中，讨论了E-n代数结构的阻碍理论，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 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.