8 problems

Let $T$ be a very presentable geometric $n$-stack, and let $X$ be a smooth projective variety. Write $X_{DR}$ for its de Rham formal stack. The de Rham mapping-stack conjecture. Th…

Let $T$ be any geometric, locally geometric, or slightly geometric $n$-stack, and let $X\to S$ be a flat projective morphism. The slightly geometric mapping-stack conjecture. The m…

Let $T$ be a geometric $n$-stack whose homotopy sheaves $\pi_i$ are vector sheaves for every $i\geq 3$, and let $X\to S$ be the morphism under consideration. The higher-vector-homo…

Let $S$ be a base scheme, let $X\to S$ be a flat projective morphism, and let $K({\bf G}_m,2)$ denote the Eilenberg–Mac Lane stack. For any group scheme $G$ of finite type, let…

Let $T$ be a geometric $n$-stack that is very presentable in the sense that its fundamental groups over artinian bases are affine and its higher homotopy groups are vector sheaves.…

Category-theoretic and quiver-bundle moduli stack equivalence conjecture. There is an equivalence of stacks

Compatibility conjecture. This isomorphism is compatible with the $(n-2)$-shifted symplectic structures on both sides. The mapping stack $\mathrm{Map}(C_{\mathrm{dR}},X)$ is known…

Nonabelian mixed Hodge structure conjecture. There is a non-abelian mixed Hodge structure on