22 problems

Let $\mathcal{O}$ be a nice ring, let $\ell$ be prime, and let $\mathbb{S}/\ell$ denote the mod-$\ell$ sphere spectrum. Let $L_1^f$ denote finite-height localization, let $v_1$ be…

A spectrum is $K(n)_*$-acyclic when its Morava $K$-theory homology $K(n)_*$ vanishes, and it is $T(n)_*$-acyclic when its telescope homology $T(n)_*$ vanishes. The Telescope Conjec…

For $n\geq 0$ and $p>2$, let $A_*=\operatorname{BP}\langle n\rangle_*[x]/x^e$. Let $w_{n+1,i}$, $y_{j,i}$, and $\gamma_{p^m}(\sigma^2x^e)$ denote the classes introduced in the prec…

Let $R$ be a ring spectrum of height $n$, meaning that $K(n)_*R\neq 0$ and $K(m)_*R=0$ for every $m>n$, where $K(m)$ denotes Morava K-theory. Height red-shift conjecture. The algeb…

Fix a prime $p$. Let $R$ be a suitable $\mathbb{E}_1$-ring of fp-type $n$, where fp-type is the chromatic finiteness notion described in the source, and let…

Let $BP\mathbb{R}^{HHR}$ denote the non-equivariant spectrum conjectured by Hill, Hopkins, and Ravenel, and let the spectra under discussion be the completed Johnson–Wilson-type sp…

Franke's algebraicity conjecture. There should be an equivalence of plain categories

Let $K$ denote non-connective $K$-theory. Let $p$ be an odd prime, let $F$ be a number field, and let $S$ be a finite set of finite places of $F$ such that…

Let $\widehat Z$ be the telescope of the $v_2$-self-map on $Z$. Let $x_i=h_{i,0}^2$. Parabola conjecture, height 2. The differentials of the preceding height-$2$ differentials conj…

In the $\mathrm{tmf}$-Adams spectral sequence for $Z$, let $\epsilon_j,\bar\epsilon_j\in\{0,1\}$ and let $m\gg0$. Differentials conjecture, part 2. The $v_2$-families … support dif…

Let $x_i=h_{i,0}^2$ in the $\mathrm{tmf}$-Adams spectral sequence for $Z$. Hidden-extension conjecture. There are hidden extensions … This is suggested by comparing the known…

Let $x_j=h_{j,0}^2$. In the $\mathrm{tmf}$-Adams spectral sequence for $Z$, use indices $i\geq3$, $l\geq i+3$, $m<2^{i+1}-1$, $k_j\geq0$, and $\epsilon_j,\bar\epsilon_j\in\{0,1\}$.…

Consider the May–Ravenel spectral sequence used to compute the relevant $\mathrm{tmf}$-resolution input for $Z$. Torsion conjecture. The May–Ravenel spectral sequence collapses at…

Let $\widehat{Z}$ be the telescope of the $v_2$-self-map on $Z$, and let $x_i:=h_{i,0}^2$. Parabola conjecture. The localized Adams spectral sequence for $Z$ collapses at $E_5$, an…

Consider the localized Adams spectral sequence … with $E_2$-term $\mathbb{F}_2[v_2^{\pm},\widetilde{h}_{2,1},h_{3,0},h_{3,1},h_{4,0},h_{4,1},\ldots]$. Differentials conjecture. In…

Let $E_r$ denote the pages of the tmf-resolution spectral sequence for $Z$, whose $E_2$-page satisfies $\prescript{\mathrm{tmf}}{}{E}_2^{s,t}(Z)=0$ for $s>(t-s+12)/11$. Vanishing-l…

Assume $p$ is sufficiently large with respect to $n$. Let $E$ be the height-$n$ theory in the source, let $I_{n-1}$ be the corresponding invariant ideal, and let $L_{n-1}$ and…

Let $L_n$ be the algebraic localization functor at height $n$. An object $X$ is $E(n)_*$-local if, whenever $E(n)_*\otimes_{BP_*}T\simeq 0$, the space of maps…

Let $\operatorname{Stable}_{BP_*BP}$ be the stable category of $BP_*BP$-comodules. For each $n$, let $L_n^f$ denote finite localization with respect to a finite type $n$-spectrum,…

The $\log_p$-conjecture. For all $n\geq r$, one has

Arithmetic duality conjecture. There is a perfect pairing

Motivic truncation conjecture. For purely $v_1$-periodic strictly commutative ring spectra $B$ there is a spectral sequence