Mathematical Sciences: Topics in Homotopy Theory

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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.