10 problems

Let ${\mathbf{Spt}}^\Sigma_{{\mathbb P}^1}(k)$ be the category of symmetric $\mathbb{P}^1$-spectra over $k$, and let ${\operatorname{{\mathcal S}{\mathcal H}}}(k)$ be its homotopy…

Let $k$ be the base field and let ${\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$ be the Morel–Voevodsky homotopy category of $S^1$-spectra. For an integer $d$, let…

Let $F$ be a field, let $\mathrm{KGL}\oldin\mathrm{SH}(F)$ be algebraic $K$-theory, and let $\mathrm{K}(F)\oldin\mathrm{Sp}$ denote its topological realization. Let the slice filtr…

Let $Q_8$ be the quaternion group, let $C_2$ be its indicated subgroup, and let $N_{C_2}^{Q_8} MU\mathbb{R}$ denote the quaternionic norm of real bordism. For a subgroup $L$ of…

Let $G$ be a finite group and let $\mathrm{AN}^{RO}_G(\mathbb S)$ be the $RO(G)$-filtered Adams--Novikov object. Applying the totalization functor gives the filtered $G$-spectrum…

Let $X$ be an $n$-dimensional connected scheme smooth and projective over ${\bf C}$, and suppose that … is a conjectural Chow–Künneth decomposition. Let ${\bf L}$ denote the Lefsch…

Voevodsky's conjecture. The easy part asserts that there is an isomorphism

Let $k$ be a field of finite virtual $2$-cohomological dimension and finite $p$-cohomological dimension for every prime $p>2$. The $I(k)$-completed slice tower is the slice tower a…

Let $k$ be a perfect field and let ${\operatorname{{\mathcal S}{\mathcal H}}}_{\operatorname{fin}}(k)$ be the thick subcategory generated by the objects…

Pullback slice-tower conjecture. The slice tower for a $G/N$-spectrum pulls back to the slice tower for the pullback $G$-spectrum, where the $k$-slice becomes the…