106 problems

Ravenel's telescope conjecture. The functors

Ausoni–Rognes chromatic redshift conjecture. The algebraic $\mathrm{K}$-theory of an $\mathbb E_1$-ring spectrum of fp type $n$ is fp of type $n+1$.

Let $M$ denote the classical Mahowald invariant, and let $v_{n-1}$-periodic and $v_n$-periodic families be families in the classical stable homotopy groups organized by the corresp…

Let $p$) be a prime, and let $L_h$ denote chromatic localization at height $h$. The $K(n)$-local sphere is expected to exhibit a precise transchromatic decomposition under the loca…

Let $p$ be an odd prime, let $C/\overline{\mathbb{F}}_p$ be a supersingular elliptic curve, and let $Q(\ell)$ be the totalization of the semi-cosimplicial spectrum … Write…

Let $M(\alpha)$ denote the Mahowald invariant of a 2-primary stable homotopy class $\alpha$, and let $\nu_n$-periodic and $\nu_{n+1}$-periodic families be periodic families in the…

Let $X$ be a finite spectrum of type $n-1$, let $L_n$ denote localization at the $n$th monochromatic level, let $X_p^{\wedge}$ denote the $p$-completion of $X$, and let $t$ denote…

The minimality conjecture for A(n). If $n\geq 2$, then $\langle A(n)\rangle$ is a minimal nonzero Bousfield class. Moreover, every such $E$ is Bousfield equivalent to a finite wedg…

Finite-generation conjecture. The graded group

Rognes' localization conjecture. These maps should fit into a cofiber sequence in the stable category

Half-sphere conjecture. If $p>2$ and $N$ is a topological generator of $\mathbb{Z}_p^\times$, then the natural sequence

Let $R$ denote the root invariant, which assigns to an element of the stable homotopy groups of spheres a coset of elements detected in the Atiyah–Hirzebruch spectral sequence for…

Let $F(n)$ be a finite complex of type $n$ equipped with a $v_n$-self map, let $T(n)=v_n^{-1}F(n)$ be its mapping telescope, and let $L_nF(n)$ denote the corresponding localization…

Over $\mathbb{C}$, let $\sigma\oldin\pi_{7,4}^{\mathbb{C}}(\mathbb{S}^{\mathrm{top}})$ and $\overline{\sigma}\oldin\pi_{19,11}^{\mathbb{C}}(\mathbb{S})$ be the motivic classes name…

In classical stable homotopy, let $\nu\oldin\pi_3(\mathbb{S}^{\mathrm{top}})$ and $\epsilon\oldin\pi_8(\mathbb{S}^{\mathrm{top}})$ be the indicated homotopy classes, and let $v_2$…

Over $\mathbb{C}$, let $\overline{\kappa}_2\oldin\pi_{44,24}^{\mathbb{C}}(\mathbb{S})$ be the motivic class obtained from the class detecting…

Over $\mathbb{C}$, let $\mathbb{S}$ be the motivic sphere, let $\eta$ be the first Hopf map, and let $w_1^4$ and $w_2^{32}$ denote the indicated motivic periodicity self-maps. The…

At the prime $2$, a finite complex means a finite spectrum in the classical stable homotopy category, and a $v_2^8$-self-map is a self-map inducing the indicated height-two periodi…

Telescope conjecture for $fp$-spectra. The telescope conjecture holds for $fp$-spectra; equivalently, for every $fp$-spectrum $X$, the localization map

The spectral Sullivan conjecture. The connective Bousfield class of $L(k)$ is the same as that of some type $k$ finite spectrum.

The category of $K(1)$-local motives relative to $\overline{\mathbb Q}$ is the category under consideration, with cellular objects understood as those in the localizing subcategory…

Let $E_h$ be Morava $E$-theory at a prime $p$, with coefficient ring … Let $\mathbb{S}_h$ denote the Morava stabilizer group, and let $G\subseteq\mathbb{S}_h$ be a finite subgroup.…

An fp spectrum is a finite-periodic spectrum in the sense used in the paper. Mahowald–Rezk conjecture. All fp spectra satisfy the conditions of the telescope conjecture. The paper…

Let $h=2^{n-1}m$ with $m$ odd, and let … be Levy's Euler characteristic map. Levy's conjecture. When $h\not\equiv 2$ mod $4$, the image of $\chi_{BP\langle h\rangle}$ is generated…

Barthel–Carmeli–Schlank–Yanovski orientability conjecture. The category $\Mod_{\ModEn}$ is $(\SS_{(p)},n+1)$-orientable.