Mathematical Sciences: Stable Homotopy

本课题拟在代数拓扑学的两个领域进行研究：球面的稳定同伦群和同伦极限问题。在第一个问题中，研究者将研究稳定同伦群中的某些三阶周期现象。这是Miller、Revenel和Wilson提出的研究稳定同伦群的全局行为的一般程序的延续。本研究的结果将增强我们对稳定同伦的潜在周期性的认识。在第二个问题中，作者将研究所谓的“同伦极限问题”。这个问题的重要性在于，许多众所周知的猜想和定理都可以表述为特殊的同伦极限问题。目标之一是将已知的同伦极限问题的正解扩展到包括代数k理论中的其他重要空间。所得结果可应用于重要同伦理论问题的研究。基本上，同伦理论是研究所有维度的球体和其他类似的自然几何物体的拓扑结构的工具。它是一个代数工具，因为代数允许计算，所以很有用。

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.