3. Postdoctoral Program - MSRI

C 2 k as k goes to infinity (with inclusion of fixed points as the structure maps) we
get another spectral sequence computing π∗ limk T HH(Z[C2]) C 2 k .
Using calculations from our paper “RO(S 1 )-graded TR-groups of Fp, Z and
ℓ” we understand the E2-term of this spectral sequence fairly well, but we kept
getting the wrong answer in cases we can understand for different reasons. Finally
we figured out that this happens because the spectral sequence does not converge!
In retrospect that should not be too surprising, because we built T HH(Z[C2]) as
a direct limit, and we cannot expect direct limits and inverse limits to commute.
We still have a spectral sequence converging to π∗T HH(Z[C2]) C 2 k for each k, so
I am still hopeful that our method will tell us something interesting.
2.4. Algebraic K-theory in multiple variables. This is joint work with Teena
Gerhardt, Mike Hill, and Ayelet Lindenstrauss. We do have some concrete results,
but nothing is yet written up and it is still work in progress. We want to understand
K(k[x1, . . . , xn]/(x a1
1
, . . . , xan
n )), which we can do if we understand T HH of that
ring sufficiently well.
By a straightforward generalization of previous work of Hesselholt and Madsen
we can write T HH as the iterated cofiber of a certain n-cube of easier-to-understand
spectra. The key observation is that this is actually an n-cube of cyclotomic spectra,
which essentially means that the K-theory is given by taking the iterated cofiber
of a related n-cube of spectra.
If k = Fp and p does not divide any of the ai’s we can compute the K-theory
groups explicitly. Similarly, if k = Z we can compute the rationalized K-theory
groups explicitly.
Our hope is that there is some underlying structure, a generalized Witt vector
construction, which gives a unified description of the K-groups in all cases. But
that part is still work in progress.
2.5. Structured Quartet Research Ensemble. The American Institute of Mathematics
(AIM) has a program called SQuaRE which allows a small group to meet to
work on a specific program. During the fellowship period (and immediately after) I
met Andrew Blumberg, Teena Gerhardt, and Mike Hill twice at AIMs center inside
Fry’s electronics store in Palo Alto, and once in Chicago to work on “algebraic
K-groups”.
One of the problems we were interested in is that of algebraic K-theory of Thom
spectra. Given a loop map f : X → BF , where BF classifies stable spherical
fibrations, we get a ring spectrum T h(f). In previous work Andrew Blumberg and
coauthors developed a good model of T HH(T h(f)) as a spectrum. However, to
be able to approach K(T h(f)) we need to understand T HH(T h(f)) as a genuine
S 1 -spectrum.
After some early missteps, we believe we now have such a model of T HH(T h(f))
as a genuine (indeed cyclotomic) S 1 -spectrum. In the course of our work we believe
we also managed to came up with a better model of T HH of a commutative ring
spectrum as the left adjoint of the forgetful functor from commutative genuine S 1 -
spectra to commutative ring spectra. This is important because the current model
of T HH as a cyclotomic spectrum is rather complicated and not very conceptual.
Since nothing has yet been written down this is still work in progress.
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