14 problems

Bhatt–Lurie’s cohomological dimension conjecture. The functor

Let $R$ be a topologically finitely generated and $p$-complete ${\mathcal{O}}_{\mathbf{C}}$-algebra, and let … be an ${\mathcal{O}}_{\mathbf C}$-homomorphism. A rational Hodge--Tat…

Let $X$ be a smooth scheme over a field of characteristic $p$. Write $X_{\mathrm{fpqc}}$ for its fpqc site, let $T_{X/k}$ be its tangent bundle, and let $T_{X/k}^{\sharp}$ be the f…

Let $\mathrm{Stack}_{\mathrm{Bun},G,X,n}$ be the stack of $G$-bundles used in the paper, and let … and … be the indicated endomorphism-line and local subcategories. Breuil–Schneide…

Unconditional local Shimura conjecture. The natural transformation

Let $k$ be the residue field, let $\mu$ be 1-bounded, and let $\mathscr{Q}\in G\mathchar`-F\mathchar`-\mathrm{Gauge}_\mu(k)$. Let $R_{G,\mu}$ represent the deformation functor on c…

Let $R$ be a complete regular local ring over $W(k)$ with residue field $k$, let $(\mathfrak{S},(\mathcal{E}))=(W(k)[[t_1,\dotsc,t_n]],(\mathcal{E}))$ be a prism of Breuil--Kisin t…

Motivic filtration conjecture. The spectrum $\mathrm{TP}(R/X(p))$ admits a motivic filtration satisfying

Motivic filtration conjecture. There is a filtration $\mathrm{F}^\star_\mathrm{mot} \mathrm{THH}(R/J(p))$ such that

Equivalence conjecture. The functors $D_X$ and $D_X^{\operatorname{perf}}$ each define an equivalence of categories. This conjecture predicts that the comparison between the overco…

Log-prismatic extension conjecture. There exists such a morphism

Formal identity conjecture. One has

Reverse stratification conjecture. The expression

Let $(A,d)$ be a perfect prism such that $A/d$ is $p$-torsion free, and let $S$ be a smooth $A/d$-algebra. Here $W_n\Omega_{S/(A/d)}^i$ denotes the degree-$i$ de Rham–Witt group, a…