(P. Goerss) Chromatic Assembly.
Assuming we know ${L}_{K(n)}X$, give an effective procedure—or even a systematic set of examples—for computing ${L}_{n-1}{L}_{K(n)}X$; equivalently, given some way to understand the $n$th monochromatic layer ${M}_{\mathrm{nY}}$ where $Y$ is $K(n)$-local. As a point of entry, first study ${L}_{K(n-1)}{L}_{K(n)}X$.
Some preliminary algebraic work has been done in the papers of Torii (MR2004428), which uses work of Gross to study what happens to deformations of formal groups. For example, given a height $n$ formal group $G$ over an algebraic extension $\mathbb{F}$ of ${\mathbb{F}}_{p}$, we can take the deformation of $G$ to $\mathbb{F}[\phantom{\rule{-0.1667 em}{0ex}}[{u}_{n-1}]\phantom{\rule{-0.1667 em}{0ex}}]$ and then consider this deformation as a height $n-1$ formal group over $\mathbb{F}[\phantom{\rule{-0.1667 em}{0ex}}[{u}_{n-1}]\phantom{\rule{-0.1667 em}{0ex}}][{u}_{n-1}^{-1}]]$. We might examine and reinterpret this algebra in homotopy theory and then extend this to a calculation and interpretation of
where ${E}_{s}$ is Morava $E$-theory. This is surely naive. Write ${C}_{n}$ for the cyclic group. Then the formal group over $({E}_{n}{)}_{*}$ is the formal spectrum of
and we probably should consider the inverse system
which is a $p$-divisible group. Thus $p$-divisible groups appear again.
The Chromatic Splitting Conjecture asserts that the map $j:{L}_{n-1}X\to {L}_{n-1}{L}_{K(n)}X$ in the chromatic fracture square is a split inclusion—and much more besides. Can even this simple statement be verified, at least for some $X$?
Revisit the Chromatic Splitting Conjecture. If it’s true, an initial case to study would be when $n$ is very small with respect to the prime $p$. Are there cases where it might not true? For example, if $n$ is not a unit in $p$, the reduced norm map ${\mathbb{G}}_{n}\to {\mathbb{Z}}_{p}$ from the Morava stabilizer group doesn’t have a central splitting. If ${\mathbb{G}}_{n}^{1}$ is the kernel of the reduced norm, there is then a Lyndon-Serre-Hochschild Spectral Sequence
and the action of ${\mathbb{Z}}_{p}$ can be very complicated. Does that twist the cohomology so that the CSS becomes difficult? Recent work of Beaudry might be applicable.
(P. Goerss) Calculations in $K(2)$-local homotopy theory. Much has been done, but more can be understood. For example, use Olivier’s thesis (Strasbourg 2013) to understand the Shimomura school’s calculations at large primes.
(P. Goerss) Structured ring spectra $p$-divisible groups. A map $\mathcal{M}\to {\mathcal{M}}_{\mathrm{fg}}$ from an algebraic stack to the moduli stack of formal groups is realizable if there is a sheaf $\mathcal{O}$ of ${E}_{\mathrm{\infty}}$-ring spectra on $\mathcal{M}$ (in an appropriate topology) with ${\pi}_{*}\mathcal{O}\cong {\omega}^{\otimes */2}$, where $\omega $ is the sheaf of invariant differentials.
It has been asserted that it’s not possible to realize ${\mathcal{M}}_{\mathrm{fg}}$ itself, mostly because the only sensible topology is the $\mathrm{fpqc}$-topology. It would nice to have this fact proved and recorded, if it’s true.
(N. Naumann) There is an argument that it is not possible to adjoin a $p$‘th root of unity to the $p$-adic $K$-theory spectrum as an ${E}_{\mathrm{\infty}}$ ring spectrum (which seems likely to be originally due to Hopkins). You can view this as a special case of adjoining a Drinfel’d level structure to the universal deformation space of height 1. As such it generalizes to all heights, and I wrote that up in 2009 for Matt Ando.
(T. Lawson) Whether or not the realization problem can be solved in a much weaker setting (e.g. merely on the spectrum level) is also something that would be worth proving.
Results of Lurie give realization criteria for 'etale maps $\mathcal{M}\to {\mathcal{M}}_{p}(n)$ where ${\mathcal{M}}_{p}(n)$ is the moduli stack (over ${\mathbb{Z}}_{p})$ for $p$-divisible groups of height $n$ with formal part of dimension 1. Explore the geometry of the map ${\mathcal{M}}_{p}(n)\to {\mathcal{M}}_{\mathrm{fg}}$ to the moduli stack of formal groups. Understand the resulting descent problem as well. The map is not representable if $n>1$.
All ${E}_{\mathrm{\infty}}$-ring spectra come equipped with power operations. Lurie’s result and its antecedent, the Hopkins-Miller theorem for Morava $E$-theories, produce ${E}_{\mathrm{\infty}}$-ring spectra but make no mention of power operations; thus it would seem the power operations are canonically dictated by the geometry of $p$-divisible groups. Explain this. Recent work of Rezk would be a place to start; this suggests analysis of the subgroup structure of $p$-divisible groups is important.
Shimura varieties. One of the important features of the Hopkins-Miller theory of topological modular forms is that one gets sheaf of ${E}_{\mathrm{\infty}}$-ring spectra on the compactified Deligne-Mumford moduli stack of elliptic curves. Are there good compactifications of Shimura varieties that would apply to the Behrens-Lawson theory of topological automorphic forms? Work of Kai-Wen Lan might be applicable.
(P. Goerss) Gross-Hopkins duality. Brown-Comenetz duality and its variants are something of a curiosity in stable homotopy theory, but it is an insight of Hopkins that in the $K(n)$-local category it is much more like Serre-Grothendieck duality.
Flesh out that statement. Some work has been done by Hopkins himself, and more by Devinatz, but this is just the start.
If possible, compare Gross-Hopkins duality to the Poincar'e duality structure of the Morava stabilizer group. Again some work has been done by people looking at Shimomura’s calculation, most lately by Behrens. Develop a larger theory that works, if possible, at primes where there’s torsion in the Morava stabilizer group.
Think about the Behrens $Q(\ell )$ spectra in this context.
I don’t know this for a fact, but Lurie’s theory probably gives a notion of Serre-Grothendieck duality in derived algebraic geometry. Make this precise and concrete; that is, give calculations. Work of Stojanoska should point the way.
(J. Morava) A question on Morava K-theories and higher gerbes
(D. Isaksen) Compute the ${E}_{2}$-term of the Adams-Novikov spectral sequence in as large a range as possible, by hand or by machine.
(A. Salch) If $E$ is a homotopy commutative ring spectrum, then a complex orientation on $E$ is a class in ${E}^{2}(\mathrm{BU}(1))$ with an appropriate property. Suppose $G$ is some gadget of some kind that has an underlying 1-dimensional formal group law. (For example, $G$ could be 1-dimensional formal $A$-modules, for some particular commutative ring $A$.) You can ask whether there exists some space $X$ equipped with a map $\mathrm{BU}(1)\to X$ such that, for any complex orientation $c$ in ${E}^{2}(\mathrm{BU}(1))$, choices of the structure of a $G$-object on the formal group associated to $c$ are in bijection with lifts of $c$ to an element in ${E}^{2}(X)$.
The answer to this is probably “no” for most choices of $G$. But it would be good to figure out when it’s yes and when it’s no.
Then you can mimic the story of formal groups but with X in the place of BU(1).
(A. Salch) Go back to the Mahowald-Ravenel-Shick paper about their attempted counterexample to the ${v}_{2}$-telescope conjecture and explain why they couldn’t quite finish what they were doing. As a side effect this would lead to a lot of good side material. Hoping to complete what Mahowald-Ravenel-Shick left unfinished is probably too much to hope for, though.
(A. Salch) Barry Walker’s thesis on ${E}_{\mathrm{\infty}}$ complex orientations of $\mathrm{KU}$ had a lot of very good stuff. Finding a way to do any of that stuff at higher heights would be great. (Ask Matt Ando and/or Charles Rezk about this.)
(A. Salch) We now know, after Johnson-Noel, that the usual ring map $\mathrm{MU}\to \mathrm{BP}$ isn’t an ${E}_{\mathrm{\infty}}$-ring map for small primes ($p<19$ or something like that). Basterra and Mandell showed, however, that it is an ${E}_{4}$-map. It would be interesting to know, for a given prime $p$, what’s the largest integer $i$ such that the usual map $\mathrm{MU}\to \mathrm{BP}$ is an ${E}_{i}$-ring map.
(J. Wu) For which spaces X are the stable homotopy groups summands of its unstable homotopy groups?
Namely, for each ${\pi}_{n}^{s}(X)$, there exists an unstable homotopy group ${\pi}_{m}(X)$ so that ${\pi}_{n}^{s}(X)$ is a summand of ${\pi}_{m}(X)$.
Here we do not require that $m$ is the same as $n$.
Counter example: If $X$ is a sphere, the answer is no. However there are some complexes $X$ having the property that its stable homotopy groups are summands of its unstable homotopy groups.
Methodology (from what we have known) for solving the question for certain special spaces $X$: If there exists a sequence of positive integers ${n}_{k}$ with ${n}_{k}$ tending to infinity such that the loop space $\Omega {\Sigma}^{{n}_{k}}X$ is a retract of $\Omega X$, then one can see that every stable homotopy group of $X$ becomes a summand of one of its unstable homotopy groups. Technically one needs to search those retracts in the loop space $\Omega X$ given as the loops on the suspensions of $X$. This methodology works for special complexes $X$ through computations. For more general cases, it seems not easy.
The telescope conjecture?.
The generating hypothesis?.
The chromatic splitting conjecture?.