This is part of an algebraic topology problem list, maintained by Mark Hovey.

- Introduce stable homotopy theory into the world of C^*-algebras,
like Voevodsky has done in algebraic geometry. More specifically, find
a model structure on some category of C^*-algebras that is useful. Then
stabilize it somehow to find a stable homotopy category of C^*-algebras
where Kasparov's KK theory is just Hom in the category. This will
probably be the same as finding a category where K-theory of
C^*-algebras is representable. I have thought about this a
little--almost the first thing you realize is that C^*-algebras are not
closed under inverse limits, so you need pro-C^*-algebras. These have
been considered by Chris Phillips in N. C. Phillips, Inverse limits of
$C\sp *$-algebras, J. Operator Theory {\bf 19} (1988), no.~1, 159--195;
MR 90c:46090. Another thing you realize is that it is easier to map
into C^*-algebras than it is to map out of them, and so it seems likely
that any model category will be fibrantly generated instead of
cofibrantly generated. But not much else is clear to me. One of the
people who knows the most about the combination of C^*-algebras and
homotopy theory is Claude Schochet. The viewpoint I am advocating here
is shared by Jim McClure, and probably Marius Dadarlat, and they may
know more about this. In particular, C^*-algebraists often make an
assumption about their cohomology theories which basically forces them
to be K-theory--I think it is Bott periodicity. To do what I am saying,
you would have to drop this assumption.

- Neil Strickland points out that several different moduli spaces are
used in differential geometry and physics. For example, there is the
moduli space people are always talking about in every talk on gauge
theory I have ever been to. I can't remember what this moduli space is
now, unfortunately--something to do with connections and the gauge
group? Anyway, the geometers tend to think only about the rational
homology of these spaces. What about their Morava K-theory? (Neil also
mentioned solitons and Seiberg-Witten theory).

- Investigate Voevodsky's stable homotopy category of schemes from a
homotopy theorist's point of view. This is obviously a huge,
unstructured problem, but I think there is room for a serious algebraic
topologist to make some inroads. In particular, Bousfield localization
is really a great thing, and they have not exploited this at all so far,
I believe. The drawback is having to learn so much algebraic geometry
before you have a chance.

- Stefan Stolz showed that a simply connected Spin manifold of
dimension at least 5 admits a metric of positive scalar curvature if and
only if its image under the orientation MSpin --> KO is 0. But the
situation is not completely understand when the manifolds are not simply
connected. It seems to get involved with the Novikov conjecture. Oh
yes, I remember--it is actually false for some fundamental groups, but
the trouble appears to be all with products with the Bott manifold.
That is, the general conjecture is: suppose M is a Spin manfold of
dimension at least 5. Then a product of M with some finite number of
Bott manifolds admits a metric of positive scalar curvature.

- Try to carry out Stolz's plan for metrics of positive Ricci
curvature. Here we expect the obstruction to lie in elliptic cohomology
rather than K-theory, and the manifolds should be MO8 manifolds. But
hardly anything is known here, I think.

- Improve on Benson-Carlson-Rickard. Recall their theorem: if G is a
finite p-group and k is an algebraically closed field, then thick
subcategories in the stable k[G]-module category (= category obtained by
killing projectives = injectives = frees) are in 1-1 correspondence with
subsets of Proj H^*(G,k) closed under specialization. There are several
open questions related to this. The most obvious one is to remove the
algebraically closed requirement. This may involve understanding some
Galois theory of stable homotopy categories, but may not be too
difficult. I sincerely hope that anybody who wants to work on the
stable module category will make the effort to understand my work with
Palmieri and Strickland on this; our setup makes things considerably
simpler, in my opinion.

- Classify the localizing subcategories of the stable k[G]-module
category. These should be in 1-1 correspondence with arbitrary subsets
of Proj H^*(G,k). I have no idea how to do this one.

- Extend the results of Benson-Carlson-Rickard to connected,
cocommutative Hopf algebras over a field, like A(n). Hovey-Palmieri
have achieved some partial success here; we reduce the calculation of
thick subcategories to the case when the Hopf algebra is
quasi-elementary. For k[G], quasi-elementary subHopf algebras look like
k[E] for elementary abelian p-groups E, and the group cohomology is a
polynomial algebra, so Benson-Carlson-Rickard use algebraic geometry to
finish the proof (this is where they require the field to be
algebraically closed). For A(n), the quasi-elementary subHopf algebras
are also exterior, but the Ext groups are bigraded. We don't know how
to do bigraded algebraic geometry, so we are stuck there. But I think
something should be doable here. Anyway, here is the abstract of the Hovey-Palmieri
paper , and here is the dvi file.

Department of Mathematics

Wesleyan University

mhovey@wesleyan.edu