As for Arakelov motivic cohomology, my general feeling is that what we did is not as interesting as the more recent stuff of Bunke and Tamme, or the research program of Baptiste Morin on Weil-etale cohomology, which paints a bigger picture than the more narrow story of Arakelov motivic cohomology. Would strongly recommend the articles of Morin on his webpage to anyone interested.

From my point of view, whenever you want to incorporate arithmetic information into anything else, you run into the problem that we seem to lack the basic tools for making sense of arithmetic schemes and associated cohomological or homotopical invariants. Here are some vague and open questions/remarks, and a lot more could be said!

1. The category of arithmetic schemes (say regular schemes of finite type over Spec(Z)) is difficult to work with, and it might not be the right category to consider at all! Could there be some other category which is nicer? There is lots of speculation here, mostly motivated by the study of zeta functions (motives, Galois representations, number-theoretic Langlands, Riemann hypothesis, special value conjectures) but not much else. For example, one should probably incorporate some kind of "arithmetic stacks" into the picture.

2. A key problem, whatever you try to do, is that for the arithmetic theory to make sense, we'd need a good understanding of finiteness properties of our objects. All the time you run into situations where you want to use some cohomology theory (motivic or otherwise) where the coefficients are G_m or some object built out of G_m, like motivic complexes. However, G_m is not constructible/compact in, say, the category of etale sheaves on a fixed arithmetic scheme X, which means that we don't know whether the cohomology groups are finitely generated, and without finiteness statements of this type, much of higher-dimensional arithmetic geometry breaks down completely. Question: Have you encountered questions related to finiteness conditions somewhere in your part of the mathematical universe? I asked a question related to this a long time ago here:

http://mathoverflow.net/questions/689/finiteness-conditions-on-simplicial-sheaves-presheaves

3. If you read about the deep ideas of Christophe Deninger and Baptiste Morin, you see formulations of a dream in which there should be a cohomological picture explaining two of the great themes in the study of zeta functions, namely special values (like the Birch-Swinnerton-Dyer conjecture) and zeroes (as in the Riemann hypothesis). One could also ask about a cohomological explanation for a third theme, namely functional equations. In all cases, we don't really have any idea of what's going on, except maybe that the Weil-etale cohomology of Morin et.al. should eventually develop into the right thing for the first issue. However, it is very natural to think that underlying this picture one should have some higher-categorical invariants or structures. You see some traces of this in the work of Morin and Flach, in which they work with classical topoi only. You also see something of this in our Arakelov paper where we work with motivic homotopy theory to make some constructions. However, motivic homotopy theory cannot be the right setting in general because in arithmetic we don't want to invert the affine line. ]]>

Schouten-Nijenhuis bracket for derivations

with out invoking vector fields ]]>

brief note on *deformation quantization of the 2-sphere*

(to go along with the existing *geometric quantization of the 2-sphere*)