136 problems

Kollár's conjecture. If

Let $(X,d,\mathfrak{m})$ be an $RCD^{\ast}(0,N)$ space, where $RCD^{\ast}(0,N)$ denotes the relevant synthetic non-negative Ricci-curvature condition. Milnor's conjecture. The fund…

Let $X$ be a smooth projective variety with big fundamental group and Kodaira dimension satisfying … A fundamental group is big in the sense defined in the source. Kollár's structu…

Let $k$ be a finite field of characteristic $\ell\ne p$, let $X$ be a geometrically connected smooth curve over $k$, and let … be the exact sequence for the arithmetic fundamental…

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$, and let $\pi_1(X)$ denote its étale fundamental group. An isocrystal on $X$ i…

Let $(X_i,d_i,\mathfrak{m}_i)$ be a sequence of $RCD^{\ast}(K,N;D)$ spaces, and let $(\widetilde{X}_i,\widetilde{d}_i,\widetilde{\mathfrak{m}}_i)$ be their universal covers. Assume…

Let $(X,d,\mathfrak{m})$ be an $RCD^{\ast}(K,N;D)$ space. If its fundamental group $\pi_1(X)$ is finite, then Fukaya–Yamaguchi conjecture. It contains an abelian subgroup $A$ of in…

Abelianity Conjecture. Every special compact Kähler manifold $X$ has an almost abelian fundamental group; that is, $\pi_1(X)$ has an abelian subgroup of finite index.

Let $S$ be an orientable surface, let $M$ be a 3-manifold, and let $f\colon S\to M$ be a map. The induced homomorphism is … Simple Loop Conjecture. If $f_*$ is not injective, then…

Let $n$ and $q$ be positive integers with $n \geq 2q$. Let $(M,g)$ be an $n$-dimensional geodesically complete semi-Riemannian manifold of index $q$, and suppose its sectional curv…

Let $S$ be a numerical Campedelli surface, meaning a minimal surface of general type with $p_g(S)=q(S)=0$ and $K_S^2=2$, and let $\pi_1(S)$ denote its topological fundamental group…

Let $Z$ be as in paragraph A16, let $X$ be the corresponding variety, and suppose that $x,y$ are $Z$-points of $X$. Let $s_\mathfrak{p}^*$ denote the specialization functor to the…

Let $G$ be a finitely generated group with $g$ generators. For a hyperbolic $3$-manifold $N$, write $\operatorname{inj}_N(x)$ for the injectivity radius at $x$, and let the convex…

Let $(Y,y)$ be a finite aspherical CW-complex which is not acyclic, and let $f\colon (X,x)\to (Y,y)$ be a continuous map. Define the degree of $f$ by letting $\deg(f)$ be the least…

Extension conjecture. The intersection form on $\pi_2M$ is extended from the integers.

Free abelian link-group conjecture. Then $L$ is welded equivalent to the unlink.

Auslander conjecture, special case. The fundamental group $\pi_1(M)$ contains a solvable subgroup of finite index.

Generalized Calabi–Markus conjecture. The fundamental group $\pi_1(M)$ is finite, and $M$ is noncompact.

Let $f$ be a singular function, let its bifurcation diagram be the relevant discriminant in the parameter space of a versal unfolding, and let the braid monodromy group be the imag…

Let $f$ and $g$ be singular functions such that $f$ is adjacent to $g$, so that a versal unfolding of $g$ is also a versal unfolding for $f$. Let $4mu$ and $tilde\mu$ be the corres…

Let $f$ be a singular function with Milnor number $4mu$, and let a Dynkin diagram of $f$ have vertex cardinality $4mu$. Write $t_1,\cdots,t_4mu$ for generators indexed by its vertic…

Finite-order translation conjecture. The characteristic variety is a union of translated subtori, with each translation given by a point of finite order.

Nielsen-type conjecture. The natural monomorphism

Let $M$ be a nef Kähler manifold of dimension $n$. Suppose there is an epimorphism … where $\Gamma$ is a torsion-free nilpotent group of rank at least $2n$. Nilpotent quotient conj…

Let $X$ be a simply connected algebraic surface, let $D\subset\mathbb{CP}^2$ be the branch curve of the relevant projection, and let $G^0$, $K$, $\bar{G}$, $\phi$, $\Lambda$, and…