Abstract:
In the past three decades, homotopy theorists have found increasingly deep
connections between stable homotopy theory and algebraic geometry. This series of talks
will outline the necessary background and describe some recent progress that is joint
work with Mike Hill and Mike Hopkins. Here is a brief description. The best approach to
the stable homotopy groups of spheres is the Adams-Novikov spectral sequence. This is
the Adams spectral sequence based on complex cobordism theory (MU) or equivalently (after
localizing at a prime p) Brown-Peterson theory (BP). The relevant algebra is controlled
by the theory of 1-dimensional formal group laws. This connection was first discovered
by Quillen in 1969, then explored more deeply by Morava in the early 1970s. This was
followed by the discovery of the chromatic filtration of the stable homotopy category in
the 1980s and the work of Hopkins and Miller in the 1990s.
A formal group law over an algebraically closed field in characteristic p is determined
up to isomorphism by an invariant called the height, which is a positive integer n. A
height n formal group law has an automorphism group S_n known as the Morava stabilizer
group. It is a pro-p-group with interesting arithmetic properties. We now know that it
has a canonical action on a certain E_\infty ring spectrum E_n which is difficult to
describe explicitly, but very valuable to know. Knowing the cohomology of this action
would tell us a lot about stable homotopy, but the problem is prohibitively difficult
for n>2.
A more tractable problem involves finite subgroups G of S_n, which have been classified
by Hewett. The most interesting cases are ones with order divisible by p, which occur
when n is divisible by p-1. In these cases we can look at the homotopy fixed point set
of G acting on E_n. Studying this in depth requires some techniques from equivariant
stable homotopy theory, which we will introduce as needed.
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