68 problems

For $n\ge 2$, let $f\in C^\infty(\mathbb{R}^n)$ be locally uniformly convex, so that $D^2f$ is positive definite, and suppose its graph is complete with respect to the induced Eucl…

Let $S$ be a del Pezzo surface of anticanonical degree at most $2$ with at worst DuVal singularities. The affine Fano variety $\mathbb{P}\setminus S$ is the complement of $S$ in th…

A compact complete affine manifold is a compact manifold equipped with a complete affine structure. Auslander's conjecture. The fundamental group of a compact complete affine manif…

Let $q$ be a prime power, let $m$ be a positive integer, and identify $\mathbb{F}_{q^m}$ with an $m$-dimensional vector space over $\mathbb{F}_q$. Let $C=\{A_1,\ldots,A_m\}$ be a s…

Kalai's affine-type reconstruction conjecture. If $P$ has no missing $(d-1)$-faces, then its graph and the space of affine $2$-stresses determine $P$ up to affine equivalence. More…

Let $X\to \mathbb{A}^m$ be an $\mathbb{A}^n$-fibration, meaning that every fibre is isomorphic to $\mathbb{A}^n$. Dolgachev-Weisfeiler conjecture. The fibration is isomorphic to th…

SYZ conjecture. There is an affine manifold with singularities $B$ and maps $\phi:X\rightarrow B$ and $\check{\phi}:\check{X}\rightarrow B$ which are dual special Lagrangian fibrat…

Affine-vector-space minimal-size conjecture. The minimal size of a tight irreducible affine vector space partition of length $n$ is

Let $X$ be an irreducible affine variety. For locally nilpotent derivations $\partial_1,\ldots,\partial_k$ of ${\mathbb K}[X]$, let … A Lie-algebra closure conjecture asserts that…

Fradelizi--Meyer's conjecture. For every convex body $K\subset\mathbb{R}^n$,

Let ${\cal F}_v$ be a leaf indexed by a positive vector $v$ in the K3 period description, let $\Gamma_v$ be the corresponding stabilizer group, and let $Aut_{{\bf Z}PL,v}(S^2)$ be…

Let $B$ be a compact space with a ${\bf Z}$-affine structure with singularities, with actual and potential singular sets equal, $B^{sing}=B^{pre-sing}$, and assume this set has cod…

Assume that the singular set of the limiting space has a stratification $B^{sing}=B^{sing}_{n-1}\cup B^{sing}_{\le n-2}$, where $B^{sing}_{n-1}$ consists of the $(n-1)$-dimensional…

Let ${X}^{an}$ be the smooth analytic space associated with the Calabi–Yau variety over $K={{\bf C}}((t))$, and let $\pi_{mer}$ map meromorphic points to their limiting points in t…

Let a dual pair of Calabi–Yau families have Gromov–Hausdorff limits, and let the smooth parts of these limits carry the induced integral Monge–Ampère structures. A Monge–Ampère man…

The Calabi conjecture. There is a unique bi-polyhedral Kähler affine structure on $\Sigma\backslash D$ of type $(\lambda,\nu)$ such that the metric is Monge–Ampère:

Let ${{\cal X}}_{mer}$ be a maximally degenerate Calabi–Yau family at $q=0$, let $X_q^{new}$ be the rescaled family, and let $n$ be the complex dimension. Maximal-degeneration limi…

Work over an algebraically closed field $\mathbb K$ of characteristic zero. Let $Y$ be an affine variety, and let $\mathbb{G}_{\rm a}$ and $\mathbb{G}_{\rm m}$ denote the additive…

Let $d\geqslant 1$, and let $\lambda^*(d,1)$ denote the optimal proportion of affine $d$-flats with intersection size $1$ in the paper's extremal problem. The one-point asymptotic…

Let $d\geqslant 1$ and let $s=j\cdot 2^k$, where $j$ is odd and $1\leqslant k\leqslant d$. The quantities $\lambda^*(d,s)$ measure the optimal proportion of affine $d$-flats having…

Let $H\subset\mathbb{A}^n$ be the hypersurface defined by a polynomial $F\in\mathbb{Z}[X_1,\ldots,X_n]$ of degree $d\geq 2$, with absolutely irreducible leading form. Let…

Flexibility conjecture. $X$ is flexible.

Let $n$ be a positive integer, and let $f:{\mathbb{C}}^n\rightarrow {\mathbb{C}}^n$ be a polynomial map with Jacobian $J(f)=1$. Suppose that, for each $i$, $f_i=x_i-h_i$, where…

Let $k$ be a field, let $n$ be a positive integer, and let $f\in\operatorname{End}(\mathbb{A}^n_k)$. Its dynamical degree is … where $\deg(f)$ is the maximum of the degrees of the…

Let $\mathbb{F}$ be one of $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, let $p=\dim_{\mathbb{R}}\mathbb{F}$, and let $K\subset \mathbb{F}^n$ be a convex body with non-empty interio…