The Slice theorem and the Reduction theorem

by Christian Carrick

Tuesday Feb 24, 2026, 2:15 PM → 4:00 PM Europe/Berlin

Endenicher Allee 60/0-008 (Mathezentrum)

I will give six lectures on the Hill—Hopkins—Ravenel solution to
the Kervaire invariant one problem. I will attempt to emphasize two key
points that don’t feature prominently in their article. The first is the
study of higher real K-theories: these are the fixed points of Morava
E-theories with respect to finite subgroups of the Morava stabilizer
group. Hill—Hopkins—Ravenel set out to understand the homotopy fixed point
spectral sequences (HFPSS) of higher real K-theories, and they proved a
number of results on the action of finite groups on Lubin—Tate spaces and
on differentials in the HFPSS that have not appeared fully in print. These
results beautifully inspired their detour through Real-oriented homotopy
theory, which removed the need for higher real K-theories in their
solution altogether.

The second point concerns the technical foundations of genuine equivariant
homotopy theory needed in their solution. The work of Hill—Hopkins—Ravenel
inspired a renaissance in the foundations of equivariant and parametrized
homotopy theory, and their norm functor gave rise to the study of
parametrized monoidal structures. I will attempt to revisit their work
from the point of view of these developments. I will finish the course by
discussing developments in Real-oriented homotopy theory following
Hill--Hopkins--Ravenel. Their detour through Real-oriented homotopy can be
seen as providing good connective models of higher real K-theories, and
the study of these connective models - especially through the slice
spectral sequence of Hill--Hopkins--Ravenel - has brought about an
explosion of progress in chromatic homotopy theory at the prime 2. I will
discuss this progress as well as future directions and possibilities at
odd primes.