22 problems

Let $Sp^{\Sigma}$ be the category of symmetric spectra and let $Comm(Sp^{\Sigma})$ be the model category of commutative ring spectra. Let $LSp^{\Sigma}$ be the tensor Segal categor…

Let $Sp^{\Sigma}$ be the category of symmetric spectra with its positive model structure, and let $Comm(Sp^{\Sigma})$ be the model category of commutative ring spectra. Write…

Let $A$ be an $R$-algebra. Write $F_{R\text{-alg}}(A,A)$ for the monoid of $R$-algebra self-maps, $B F_{R\text{-alg}}(A,A)$ for its classifying space, and ${\bf Der}_R(A,A)$ for th…

Let $R$ be a commutative ring spectrum, let $A$ and $B$ be $R$-algebras, and let $f:A\rightarrow B$ be an $R$-algebra map. Write $F_{R\text{-alg}}(A,B)$ for the space of $R$-algebr…

For $p=2$, let $K(1)$ be the spectrum considered as an algebra over the completed connective complex $K$-theory spectrum $\widehat{KU}_2$, and let an opposite algebra structure mea…

Let $A$ and $B$ be differential graded algebras (DGAs), and let $\MU$ denote the complex cobordism spectrum. The MU-algebra equivalence conjecture. There exist DGAs $A$ and $B$ tha…

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 $F \colon \mathcal{J} \to \mathcal{S}$ be lax symmetric monoidal, and let $\Theta^1F$ denote its first derivative spectrum. First derivative multiplication conjecture. The spec…

Cotwist conjecture. There is a commutative diagram in $\operatorname{LinCat}_R$ identifying $T_{\operatorname{Fun}(S^n,\operatorname{RMod}_R)}$ under the equivalence…

Let $R$ be a discrete ring. Write $Q(R)$ for the spectrum associated with MacLane's $Q$-construction, let $\mathbb{Z}$ denote the Eilenberg–Mac Lane spectrum of the integers, and l…

Agreement conjecture. The non-connective $K$-theory functor $K^B$ agrees with the non-connective $K$-theory functor of Blumberg, Gepner, and Tabuada.

Let $R$ be a ring spectrum, let $v\in \pi_*R$ be a homogeneous element, and consider the localization map … Here $R[v^{-1}]$ denotes the localization obtained by inverting $v$, and…

Relative THH conjecture. The relative topological Hochschild homology spectrum should satisfy

Let $C_2$ be the cyclic group of order two, and let $\mathrm{tmf}_{C_2}$ be the homotopy pullback spectrum defined by the Tate diagram described above. A $C_2$-ring spectrum means…

May's conjecture. Then $R$ is $K(n)$-acyclic for all $n>0$.

Let $R$ be a commutative ring spectrum with telescopic complexity $n$, meaning that the thick subcategory determining its telescopic height is $\mathcal{T}_R=\mathfrak{C}_{n-1}$. H…

Spectrum-level splitting conjecture. As a spectrum, $Mj_2$ is a wedge of $k\mathrm{O}$-module spectra, $Mj_3$ is a wedge of $\mathrm{tmf}$-module spectra and $Mj^{\mathrm{c}}$ is a…

Epimorphism conjecture. A $2$-local characteristic morphism

Characteristic morphism conjecture. Each of the three displayed maps is a characteristic morphism.

Characteristic conjecture. $S/\!/2^r,\eta$ is a characteristic for $H\mathbb{Z}/2^r$.

Odd-primary characteristic conjecture. $S/\!/p^r,\alpha_1$ is a characteristic for $H\mathbb{Z}/p^r$.

Let a category have a lax or partial lax monoidal product whose structure maps are weak equivalences, and let its unit have a derived endomorphism ring spectrum. Lax monoidal produ…