My livejournal account has not yet been discontinued, despite my nearly 2 year hiatus. (I think my gmail account has!) This warrants new entries. Unfortunately the math will not be as restrained, although not quite as evident as in Ian's blog.
A geometer friend Ben from college is visiting. He told me about a process through which you can associate a curve to a knot. One can then ask if infinitely many curves of genus 0 show up, etc. Apparently to do this, one starts with a knot, look at the complement in S3, give this a hyperbolic structure, look at the associated group in PSL2C, and take the character variety. There is a distinguished component, and it is 1-dimensional. Smooth this and you get a smooth proper curve defined over a number field from your knot. My first response to this was to consider the analogy between 3-manifolds and number fields. By this analogy, one would have a process which associates to each rational prime a smooth proper curve over some number field. I would be curious to know what this is. Then one could guess what the appropriate answers are to questions about the curves.
In relation to something else he brought up, the question arose if the rank of an elliptic curve over a number field is independent of the number field. That is, is it stable under extension. This is a pretty stupid question, but I don't know the answer. Another question: given an elliptic curve E over a number field K, can the set of points over K be dense in E.
Another Ben told me while ago that étale K-theory is just the étale sheafification of algebraic K-theory. Although this should've been obvious to me, it wasn't. Algebraic K-theory gives a presheaf of spectra on the small étale site of a scheme X, and one should sheafify to get a sheaf of spectra. The global sections of this of this on X is the étale K-theory spectrum. One can make sense of the sheafification with model categories and Bousfield localization. Ben suggested that the theory of Kan extensions means you can make sense of sheafification whenever you have a theory of homotopy limits. Unfortunately, I don't know anything about Kan extensions.
A more naïve approach to sheafification might be solved as follows. For a “triangulated category” D, there should be a notion of presheaves and sheaves with values in D. These should also be “triangulated categories.” One should have notions of exactness and localization. An example of both should be given by sheafification. One should have an exact global sections functor from both presheaves and sheaves to D. There should be a descent spectral sequence, etc.
Unfortunately, trying to do this naïvely with triangulated categories fails due to the nonuniqueness of the cone. One should use DG-categories, or better A∞ categories.
This perhaps explains why one has to go through the model category structure above when D is the category of spectra, and likewise why Thomason has to work so hard to work with appropriate models of the derived category.
One can likewise introduce algebraic K-theory this way. Define it first for affine schemes, and then sheafify in the Zariski topology. I should write this all up soon.
Another example of this, would be with D the derived category of abelian groups. Then sheaves with values in D would essentially be the derived category of sheaves. The global sections functor is the usual one, and the descent spectral sequence specializes to the hypercohomology spectral sequence.