(for todd)

i think that you noticed that one of the first items on my wish list was "galois shapeshifters", so i might as well take a stab here at trying to say what i mean by that phrase... probably i don't have to bother warning that the ideas here are pretty uncertain in my mind ...

part of my experience of learning about topos theory was to hear the ritual recitation of stories about the role in its historical development of grothendieck's attempts to prove the weil conjectures...

(the ritual nature of this recitation is tied in with the "estrangement" that i've sometimes mentioned: "...an estrangement that i felt during my earlier and less successful experiences in trying to learn algebraic geometry. i’d heard rumors of something called “toposes” which according to an often-repeated story played a significant role in advances in algebraic geometry associated with alexander grothendieck and the proof of the “weil conjectures”. then i learned topos theory, especially through the inspiring (sometimes a little _too_ inspiring) lectures of bill lawvere, and although i found it beautiful and highly applicable it didn’t actually seem to help me much with learning algebraic geometry...")

so anyway, the question of "why do people make such a big deal about the weil conjectures?" has continued to bug me for a long time; and semi-recently, i've formulated a tentative answer: roughly speaking, the big deal is secretly because of what i call "galois shapeshifters" ...

roughly speaking, a "galois shapeshifter" for me is an algebraico-geometric object which lives over an algebraic number field k and is therefore subject to the action of the absolute galois group over k, and which "changes its shape" under that action. in particular, the plain ordinary homotopy type of the space of complex points of a projective algebraic variety (with the subspace topology relative to the usual topology on the ambient complex projective space) may change.

apparently serre in particular made somewhat of a project of making examples of this phenomenon as nice and explicit as possible ... explicit, dramatic, simple, low-dimensional, palpable, picturesque, etc ... i haven't gotten to the point of actually really understanding any of serre's examples yet, but it's somewhat of a project of my own, not very developed so far, to understand serre's examples, or more likely to find my own such examples and understand those...

the point of such examples, insofar as i understand it, is to develop a sense of knowing what you're up against when you try to develop functorial processes for extracting invariant objects from algebraic varieties defined over algebraic number fields which specialize in certain cases to usual homotopy-invariants of subspaces of complex projective space... which, i have the vague impression, is part of what the weil conjectures (or at least certain ways of trying to prove them) are all about ...

(again, as usual, i could easily have some crucial details mangled here, but i think that the actual situation is vaguely similar to what i'm trying to describe here ...)

thus there's a tension between the change that the homotopy type may undergo and the nevertheless apparently somewhat powerful invariants (like "etale cohomology groups" ... or something like that ...) that resist that change ... which makes me uncertain and curious as to how dramatic the examples of shapeshifters can be, and to what extent you can sense the intrinsic aspect that doesn't shift ...

it seems to me a reasonable (if unfortunate...) rule of thumb that when the word "cohomology" appears in a mathematical exposition that's the place where it changes from exposition to obscurantism conveying only the author's having given up on finding a conceptual understanding of what they're talking about... a partial exception in the case of actual cohomology groups of reasonable spaces, as opposed to the more obscure sorts of "cohomology groups" that i associate with "homological algebra" ... i guess that i hope that a better understanding of the "galois shapeshifter" phenomenon might help me in trying to develop a conceptual understanding of some things for which a conceptual understanding usually seems to be lacking ...

at some point i should try to read the n-lab article on "cohomology", which i'm imagining was written by urs schreiber ... i'm undecided so far as to whether there might be an interesting conceptual understanding buried in that article... i think that i sometimes learn interesting things from urs, though maybe always second-hand rather than directly, so far... i've noticed that he generally doesn't seem to get what i'm saying, judging from the way he usually tries to interpret it in terms of things to which it seems to be essentially unrelated ...