15 problems

Compactification conjecture. Every $m$-dimensional subset of $\mathbb{R}^n$ has an $m$-dimensional compactification that can be embedded into $\mathbb{R}^n$.

Let $K$ be a countable CW complex, and let $AE(X)$ mean that every map from a closed subset of $X$ to $K$ extends over $X$. A metrizable compactification $\widetilde{X}$ is a compa…

Griffiths conjecture. The period image $\mathlcal P$ is quasi-projective and $\varphi$ is induced by an algebraic morphism; $L$ is ample on $\mathlcal P$; and there exists a Baily–…

Relative compactification conjecture. For some $n_0\in{\mathbf N}$, $P_{n_0}$ is a relative compactification of ${\mathcal G}$ and

Locality conjecture. There exists a $G(\sF)\times G(\sF)$-equivariant sheaf ${\mathbf S}_{\rho}$ of complex vector spaces on $\overline{G}(\sF)$ such that…

A weak K-moduli space is an Artin stack when continuous isotropy is allowed. Consider such a stack parametrizing polarized irreducible holomorphic symplectic varieties, polarized a…

Let $\overline{\mathcal{Y}}_{n,k}$ and $\overline{\mathcal{Z}}_{n,k}$ be the compactifications of $\mathcal{Y}_{n,k}$ and $\mathcal{Z}_{n,k}$ described in the source, let…

Let $G$ be a subgroup of a compact group. A space is weakly pseudocompact if it is $G_\delta$-dense in some compact space, meaning that every non-empty $G_\delta$-subset of that co…

Let $\mathcal{SD}^{\mathrm{u}}$ be the space of unoriented string diagrams, let $\mathcal{SD}^{\mathrm{u}}/\sim$ be its quotient by the stated equivalence relation, and let…

Let ${\rm M}_{Dol}(S,C_{\cdot})$ be the Dolbeault (Hitchin) moduli space and ${\rm M}_B(S,C_{\cdot})$ the Betti character variety. Let $N^{\ast}_{Dol}$ and $N^{\ast}_B$ be neighbor…

Let $M_B$ be the character variety with a compactification whose boundary is a normal-crossings divisor, and let $M_{Dol}$ be the Dolbeault moduli space with Hitchin base of comple…

Let $\bar X$ be a primitive algebraic compactification of $\mathbb{C}^2$ formed by minimally resolving the singularities of a curve-germ at a point on the line $L_\infty$ at infini…

Boundary-component conjecture. The locus

For each genus $g$, let $I^{(g)}\subset\mathcal A_g$ be the locus of principally polarized abelian varieties whose theta divisor is singular at an odd two-torsion point, and let…

Intersection-number vanishing conjecture. The intersection number