Mathematical Sciences: Topics in Homotopy Theory

戴维斯将研究同伦理论的三个一般领域的问题。i)一个主要的目标是证明，在收敛于球的2初级稳定同伦群的Adams谱序列中，上面只有J像存在的直线斜率为1/5，甚至可能是1/6，而不是已经建立的3/10。这个结果的奇怪的初级模拟也将被考虑。在不稳定同伦理论中，由同一思想产生的项目包括确定特殊正交群SO(n)的v-一周期同伦群，确定SO的不稳定Adams谱序列，以及确定球的不稳定Adams谱序列中v-零扭转的更好界。ii)他希望研究发育不良实射影空间的粉碎积的稳定同伦类型的分类。iii)他将通过首先尝试阿贝尔情况来研究Landweber关于有限群的复边界与其秩的猜想。这些问题都需要同伦群计算方面的专业知识，但计算主要是人工而不是电子的，只有少量和偶尔的机器帮助。由经验丰富的直觉引导的灵活分支有望超越蛮力。这些计算的结果被设想为几何应用，这将为拓扑学家的使用提供新的工具。

Davis will work on problems in three general areas of homotopy theory. i) A primary goal is to prove that, in the Adams spectral sequence converging to the 2-primary stable homotopy groups of spheres, the line above which only the image of J survives has slope 1/5, or perhaps even 1/6, rather than the 3/10 already established. The odd primary analogue of this result will also be considered. Projects in unstable homotopy theory arising from the same circle of ideas include determining the v-one-periodic homotopy groups of the special orthogonal groups SO(n), the unstable Adams spectral sequence for SO, and a better bound on v-zero-torsion in the unstable Adams spectral sequence for spheres. ii) He hopes to study the classification of the stable homotopy types of smash products of stunted real projective spaces. iii) He will study a conjecture of Landweber relating the complex bordism of a finite group to its rank, by first attempting the abelian case. These problems all call upon expertise in computing with homotopy groups, but the computing will be mainly human rather than electronic, with only modest and occasional machine assistance. Flexible branching guided by seasoned intuition is expected to surpass brute force. Geometric applications are envisioned for the results of these calculations, which will provide new tools for the use of topologists.