Mathematical Sciences: Stable Homotopy

将在以下两个代数拓扑学领域进行研究：球面的稳定同伦群和同伦极限问题。在第一个问题中，研究者将研究球面稳定同伦群中的某些三阶周期现象。这是由Miller，Revenel和Wilson提出的研究球面稳定同伦群的整体行为的一般程序的继续。这一研究结果将加深我们对稳定同伦的内在周期性的理解。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，各种著名的猜想和定理都可以表示为特殊的同伦极限问题。其中一个目标是将同伦极限问题的已知正解推广到代数K-理论中的其他重要空间。所得结果可用于研究重要的同伦理论问题。基本上，同伦理论是一种工具，用于研究所有维度中的球体和其他类似的自然几何对象的拓扑。它是一种代数工具，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.