Abstracts for Conference on "New Perspectives on Stable Homotopy and Beyond"

In their recent breakthroughs, Barthel--Schlank--Stapleton--Weinstein used $p$-adic arithmetic geometry to compute both the rationalization of the $\mathrm{K}(n)$-local sphere and the algebraic Picard group of the $\mathrm{K}(n)$-local category. The latter is topologically generated by the $\mathrm{K}(n)$-local ordinary and determinant spheres. 

I will present some results on $KO^*(CP^n)$, $EO_{p-1}^*(CP^n)$ for odd primes $p$, using homotopy fixed point spectral sequence. Using these results we will discuss the existence of exotic structures on complex projective spaces, existence of smooth $S^1$, $S^3$ actions on exotic spheres, existence of complex hyperbolic manifolds with certain curvature properties.

A celebrated theorem of Snaith shows that complex K-theory can be obtained from the unreduced suspension spectrum $\mathbb C P^\infty$ by inverting the Bott class. A contemporary reformulation of this result is that complex K-theory classifies orientations of the multiplicative group in derived algebraic geometry.

The geometry of formal group laws informs and controls much of the structure of stable homotopy theory, and the notion of "p-typical formal group law" leads to one construction of the important Brown--Peterson spectra. In equivariant stable homotopy theory, it is not totally clear which notion of equivariant p-typicality should be used to construct equivariant Brown--Peterson spectra.