Mathematical Sciences: Stable Homotopy

研究项目如下 代数拓扑学的两个领域： 球和同伦极限问题。 在第一个问题中， 研究人员将研究某些三阶周期性 现象在稳定的同伦群体的领域。 这是一 继续研究全球气候变化的一般方案 球的稳定同伦群的行为，由 米勒，瑞弗内尔和威尔逊。 从这项研究中获得的结果 将增强我们对潜在周期性的理解， 稳定同伦 在第二个问题中，笔者将研究 所谓的“同伦极限问题”“这一点的重要性 问题在于，各种众所周知的理论和 定理可以表述为特殊的同伦极限问题。 目标之一是将已知的肯定性解决方案扩展到 同伦限制问题，包括其他重要空间，从 代数K理论 研究结果可应用于该研究 重要的同伦理论问题 基本上，同伦理论是一种研究 拓扑领域在所有方面和其他这样的自然 几何物体 它是一个代数工具，有用的，因为 代数允许计算。

The research projects to be undertaken are in the following two areas of algebraic topology: the stable homotopy groups of spheres, and the homotopy limit problem. In the first problem, the investigator will study certain third-order periodic phenomena in the stable homotopy groups of spheres. This is a continuation of a general program for studying the global behavior of stable homotopy groups of spheres, as formulated by Miller, Revenel and Wilson. Results obtained from this research will enhance our understanding of the underlying periodicity of stable homotopy. In the second problem, the author will study the so called "homotopy limit problem." The importance of this problem lies in the fact that various well-known conjectures and theorems can be formulated as particular homotopy limit problems. One of the goals is to extend known affirmative solutions to the homotopy limit problem to include other important spaces from algebraic K-theory. The result can them be applied to the study of important homotopy theoretic questions. Bascially, homotopy theory is a tool for investigating the topology of spheres in all dimensions and other such natural geometric objects. It is an algebraic tool, useful because algebra permits computation.