@JonathanBeardsley Sometimes I get emails from students in random parts of the world (usually India, but I assume that's just because there's 1bn people there) asking me random math questions about stuff I did a presentation in college. I tend to ignore those. Much more rarely I get emails by grad students that have read my papers and ask pertinent questions, and I tend to answer them (I don't know which of the two situations you're referring to)
of course i appreciate the shout-out, but it's pretty silly thing to get cited for. i feel like for mathematical honesty's sake, i should update that arxiv post to just read 'if you want to see this done right, see Section 3 of the ambidexterity paper'
@DenisNardin yeah, the former was more or less what i was referring to. i got an email asking me if someone could come to Seattle for 6 months and do their undegraduate thesis with me (I think on hyperplane arrangements, which i know next to nothing about), and another asking me about complex projective spaces
Yeah, it "looks like" mapping into a coalgebra I think?
Oh no, ok, this is better, it looks like mapping OUT of a coalgebra.
okay, yeah, so i dunno, i feel like this is such a convoluted way of saying (more or less) that a formal group over an E_\infty-ring is a functor from R-algebras to Spaces that lands in infinite loop spaces
but of course not QUITE just that, because of the whole "adic" thing, which makes them look infinitesimal
i'm not feeling great, so i didn't read that much of it, but he does say something to the effect of wanting to consider these things three different ways, yes?
and yeah, i think one of the reasons for the coalgebra way is to worry less about completions
right, things that "look like" Spf(C^{\vee}), i.e. the formal spectrum of the dual of a coalgebra
well cool. so i guess the point of this is to construct universal deformation ring spectra, e.g. Morava E-theory, entirely in the category of spectra? that seems important!
although there's still this nagging defect for me w/r/t K(n) not being e_\infty
@SaulGlasman my (probably incorrect) understanding is: naively, E-theory doesn't have a moduli-theoretic interpretation in DAG; this is because pi_* E_n only has a moduli-theoretic interpretation when you take into account its topology. but quotients don't really work well in the spectral setting. if you develop adic DAG/SAG carefully (as jacob does), then this is no longer an obstruction to the existence of a moduli-theoretic interpretation for E-theory
@skd yeah i'm far from an expert on this, but my understanding of the obstruction was always something like "well, we'd have to talk about pro-spectra and this is bad" or something
a cursory reading of the instruction says that 'the adic topology on E_n' can be understood either (1) as the pro-system {E_n / I^j}_j or (2) as the pro-system that's constant at E_n on objects and multiplies variously by the u_js
that's certainly true at the level of rings, and it's also true that (2) doesn't involve any quotients, so, great
@JonathanBeardsley yeah, sure does seem important. Sections 5-6 are a re-proof of the existence of E-theories, first after K(n)-localization and then in the global category. here's something i think would help me get a better handle on the tail: aside from the information known at pi_0 and the torsion claim of Section 6.3, what does L_K(n) R^un look like at heights > 1?
