Seminar

The Goodwillie tower of the identity, when specialized to  odd dimensional spheres, has many wonderful properties.  In particular, localized at a prime p, one gets a spectral sequence converging to the homotopy groups of the 2n+1 sphere which start from  the stable homotopy groups of certain spaces L(k,n).  When n=0, it has  been long conjectured that the spectral sequence collapses at E^2.

 This amounts to saying that certain non-infinite loop maps from

QL(k,0) to QL(k+1,0) assemble to give a long exact sequence in homotopy.

Meanwhile, infinite loop maps in the other direction appear in the statement of a conjecture of G. Whitehead from the late 1960's.

By calculating everything on primitives in mod p homology, I am able  to show that these two sets of maps fit together in the best way possible.

This proves the conjecture about the Goodwillie tower at  all primes (Behrens has a version when p=2), and simplifies my 1982  proof of the Whitehead Conjecture.

The Hecke algebras of type A may make an appearance.  Then again, they may not.