Research
My favorite questions in mathematics involve geometrically flavored classification problems whose solutions depend on the explicit computation of some invariant of a space.
A few spectral sequences I've been thinking about recently:
- the unstable Adams and EHP sequence
- the Atiyah–Hirzebruch spectral sequence for real projective spectra
- the equivariant slice spectral sequence for higher real K-theories
Homotopy groups of spheres
One of my favorite problems in algebraic topology is the classification of homotopy types of finite cell complexes. The key ingredient in this classification problem is explicit knowledge of homotopy groups of spheres.
This link will take you to the unstable Adams charts that I generated in a recent computer-based approach to computing some new homotopy groups of spheres. The $\mathrm{E}_2$-page was computed using the lambda algebra and Curtis algorithm. Unstable Adams differentials were deduced from stable Adams differentials using the algebraic EHP sequence and the unstable Leibniz rule.