45 problems

X(n)–telescope meet conjecture. For all $n$ and $k$,

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…

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

Let $p$ be a prime and let $\ell$ be a generator of $\mathbb{Z}_p^\times$ when $p$ is odd, or of $\mathbb{Z}_2^\times/\{\pm 1\}$ when $p=2$. Write $Q(\ell)_{K(2)}$ for the $K(2)$-l…

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…

Chromatic redshift conjecture. If $A$ has height exactly $n$, then $\operatorname{K}(A)$ is an $\mathbf{E}_\infty$-ring of height exactly $n+1$.

Let $X$ be a spectrum of fp-type $n$, and let $L_n^f$ and $L_n$ denote, respectively, finite and ordinary chromatic localization at height $n$. Mahowald-Rezk conjecture. $X$ satisf…

Let $p$ be a prime, and let a spectrum of fp-type $n$ mean a bounded below, $p$-complete fp-spectrum whose associated thick subcategory of finite spectra has type at least $n+1$. L…

Let $S/(h,v^4_{(1,0)})$ denote the indicated $C_2$-equivariant complex, let $a$ be the Euler-class element, and let $\widetilde{\bar{\kappa}}$ denote the specified equivariant lift…

Suppose that $X\in\mathrm{Sp}^{C_2}_{(2),\omega}$ is a type $(n,n-1)$ complex. Suppose further that the modified inductive procedure produces a $v_n$-self map as a Mahowald invaria…

Odd-prime transchromatic shearing conjecture. There is a shearing isomorphism

Height classification conjecture. The $C_{2^n}$-slice spectral sequence of $BP^{(C_{2^n})}\langle m\rangle$ contains all differentials in the slice spectral sequence of…

Hill–Hopkins–Ravenel transchromatic phenomenon. The slice spectral sequences of higher height theories contain isomorphism regions to the slice spectral sequences of lower height t…

Finite-defect telescope conjecture. Any spectrum with finite chromatic defect satisfies the condition of the telescope conjecture at all heights and primes.

Chromatic redshift conjecture. If $\alpha \notin M_{\mathbb{K}}(\alpha)$, then $M_{\mathbb{K}}(\alpha)$ consists entirely of $v_n$-torsion elements. If $w:\Sigma^dY\to Y$ is a powe…

Let $\mathrm{tmf}$ be the connective topological modular forms spectrum, let $\overline{\mathrm{tmf}}$ denote its cofiber in the unit decomposition, and let…

Let $G$ be a finite $p$-group, let $E$ be a $p$-complete complex-oriented spectrum with associated formal group of height $n$, and let $E^2(f)$ denote the induced map in degree-two…

Classical blue-shift phenomenon. The spectrum $t_{G}({\rm \inf}^G_{e}(E))^G$ is $v_{n-s_{G;E}}$-periodic for some positive integer $s_{G;E}$. When $s_{G;E}>n$, the $v_{n-s_{G;E}}$-p…

Let $E_n$ denote the height-$n$ Morava $E$-theory spectrum, and let $\rho{ ull}$ denote the $\rho$-localization functor. Write…

Let $p$ be a prime, let $A$ be a finite abelian $p$-group, and let $\mathcal{F}$ be a proper family of subgroups of $A$. Let $R$ be a commutative algebra regarded as a Borel-equiva…

Reduced Homological Vanishing Conjecture. For the parameters under consideration, this map is an isomorphism

Let $E_*$ be the coefficient ring of Morava $E$-theory at height $n$, let $E_*E$ be its cooperations, and let $v_0=p,v_1,\ldots,v_{n-1}$ denote the usual periodicity elements. For…

The height $n$ Telescope Conjecture. The Bousfield class of $T(n)$ agrees with the Bousfield class of $K(n)$:

Ausoni–Rognes redshift conjecture. The $p$-completed algebraic K-theory spectrum