144 problems

Let $n$ be a positive integer, and let $F\colon\mathbb C^{n}\to\mathbb C^{n}$ be a polynomial map with constant nonzero Jacobian determinant. Such a map is called a Keller map. The…

Let $M$ be a closed affine manifold whose linear holonomy lies in $\operatorname{SL}_d(\mathbb{R})$, so that $M$ has parallel volume. An affine manifold is complete if it is a quot…

Chern's conjecture. The Euler characteristic of $M$ is zero.

Strict convexity conjecture. The function

A plank $P$ is a region between two parallel hyperplanes in $\mathbb{R}^d$, and its relative width with respect to a convex body $K$ is … where $u$ is the normal of $P$. Let…

Flenner–Zaidenberg rigidity conjecture. The affine Pham–Brieskorn hypersurface $X_{a_0,\dots,a_n}$ is rigid if and only if $\min\{a_0,\dots,a_n\}\geq 2$ and at most one element $i$…

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…

Let $K\subset\mathbb R^n$ be a convex body, let $\mathfrak{P}_n$ be the set of continuous affine invariant points $p:\mathcal K^n_0\to\mathbb R^n$, and define … Also let … Here…

Liendo's FML rationality conjecture. If

Let $V(p)$ be a hypersurface in affine $n$-space, and let $R$ be its coordinate $K$-algebra. A hyperplane in affine $n$-space is isomorphic to affine $(n-1)$-space. Zariski's cance…

Maubach–Willems conjecture. There are infinitely many finite extensions $K/\mathbf{F}_q$ for which

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…

Let $\mathbf{k}$ be an algebraically closed field, and let $\operatorname{Aut}_{\mathbf{k}}(\mathbb{A}^n)$ denote the group of automorphisms of affine $n$-space over $\mathbf{k}$.…

A metric on a manifold is compatible with an affine connection when the affine connection preserves the metric under parallel transport. Weyl's conjecture. Riemannian metrics are p…

A Margulis spacetime is a quotient of Minkowski space $\mathbb{R}^{2,1}$ by a properly discontinuous affine action, and a crooked plane is a piecewise-linear surface in…

Let $(\mathbb{R}^2,\beta)$ be the real affine plane equipped with the ternary betweenness relation $\beta$, and consider expansions of this structure by arbitrary unary predicates.…

Jacobian conjecture. If $J(P,Q)\equiv c\neq 0$ is a non-zero constant, then $F$ is invertible.

Popov's conjecture. If $G$ is connected, then $\Gamma$ is cyclic.

Zaidenberg's quasihomogeneity conjecture. Up to a polynomial automorphism of $mathbb C^n$, the polynomial $p$ is equivalent to a quasihomogeneous polynomial.

Zaidenberg's isotriviality conjecture. If $X$ is contractible, then $p$ is an isotrivial polynomial: its generic fibres are pairwise isomorphic.

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 $K$ be the ground field, let $V(p)$ be a hypersurface in affine $n$-space, and call two subvarieties equivalent if an automorphism of the ambient affine space takes one onto th…