See refs to tmf and to possible generalised class field theory at http://www.math.uio.no/~rognes/cas/application.html http://www.math.uio.no/~rognes/papers/oag2005.pdf
Segal: Categories and cohomology theories
topoligal modular forms: jacob lurie constructs this spectrum, while hopkins et al only proved that it must exist
is luries machinery applicable also to motivic spectra? is there a structure of symmetric monoidal infinity category?
Possible starting points:
Rognes
Schmidt
TAF
Panin, Rondigs, …
Reading list for top mod forms: http://www.math.uiuc.edu/~ganter/talbot/index.html
Strauch idea: homotopy for rigid analytic spaces. See notes
Suppose that formal group laws arise for oriented cohomology theories. Then perhaps there is something like algebraic elliptic cohomology. Perhaps this is related to questions in noncommutative Iwasawa theory (of elliptic curves) just like K-theory (“the multiplicative FGL”) is intimately related to ordinary Iwasawa theory, and the conjectures of Vandiver, Leopoldt. For each kind of FGL, we study on one hand torsion points of the corresponding algebraic group, and the field extension these points generate, and on the other hand the oriented cohomology theory corresponding to the FGL.
Kth76 (1995): Voevodsky discusses algebraic Morava K-theory, and defines there a cohomology theory by four axioms: Exactness, Nisnevich descent, and two types of homotopy invariance.
there should be a short and powerful way to describe spectral sequences using the homotopy language, analogous to the description of Verdier as in Freitag-Kiehl
see Deligne: regularized dets, in Motives, for a cohomological approach to L-functions over number fields
Iwasawa modules: can we endow them with a model category structure so that weak equivs are the obvious things? If so, would this be of any help in noncomm Iwasawa Th?
Morel, in the little book, constructs the stable homotopy category of the category of smooth schemes over som base. This gives a hold on the cohomology theories on such smooth schemes. One thing he does is to describe the spectrum defining algebraic K-theory. Can we define the stable homotopy category for rigid geometry??? Can we speak about cohomology (K-theory?) for objects from rigid geom?
Toen has a preprint (perhaps his thesis?) on K-theory and cohomology of algebraic stacks.
Schmidt, Wiesend and perhaps others have written on class field theory and singular homology for arithmetic schemes.
Hovey: Currently we know how to do equivariant stable homotopy theory only when the structure group G is compact Lie. But I bet we can do it when the group G is profinite as well. The main example I am thinking of is the Morava stabilizer group, though one could warm up with the p-adics. Presumably Morava E-theory should be an object of this category, but how?
In a Wiki article, there is a notion of a TQFT as a functor from Bord to VS. What axioms does this satisfy? Is Bord a geom. category?
What about the issue of torsion in cohomology theories? In the Grothendieck-style approach, the philosophy seem to be to ignore torsion (tensoring with Q) (but what about the Picard group, which is finite in some cases???). However, in for example K-theory, torsion questions are central, it seems. What about axiomating theories with torsion? Can we split a theory into free and torsion part? Combinatorial approach to torsion part, via axioms? (For example, showing first uniqueness of alg K-theory and then showing that a conjectural description satisfies axioms, we might be able to prove torsion conjectures for K-theory.)
Perhaps maths should always be learned not from definitions, but from looking at how the various objects relate to each other.
nLab page on Some ideas IV