136 problems

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…

Let $(X,\omega)$ be a simply connected symplectic $4$-manifold. For sufficiently large $k$, let $G_k^0$ be the relevant subgroup associated with the branch curve of the degree-…

Simplified presentation conjecture. The presentation has the following properties:

Fix a prime $p$, and for a group or space $X$ let $\beta_i^p(X)$ denote the dimension of $H_i(X,\mathbf{Z}_p)$. A homology 4-sphere is a 4-manifold with the homology of the 4-spher…

Let $X$ be a simply connected symplectic manifold, let $G_k^0(X,\omega)$ be the reduced stabilized fundamental group, and let $L_k$ and $K_X$ be the classes appearing in the constr…

Let $X$ be a simply connected complex surface, let $f_k:X\to\mathbb{CP}^2$ be a generic projection defined by sections of $L^{\otimes k}$, let $\tilde{X}_k$ be the associated Galoi…

Let $X$ be a simply connected symplectic manifold, let $G_k^0(X,\omega)$ be the reduced stabilized fundamental group, let $\Lambda_k$ be the image of…

Let $X_K$ be a smooth connected scheme over $K$, let $x_0,x_1\in X(K)$, and let $F_{x_i}$ denote the corresponding fibre functors on the category under consideration. Write…

Deligne's conjecture. The $\mathbb Q_\ell$-form

Realisability conjecture. The bundles $Q$ and $Q'$ are equivalent if and only if $\varphi$ is realisable.

Let $X_0$ be a nontrivial chain or ring of curves of genus greater than $0$, let $g$ be its arithmetic genus, and let $X_I$ be the family of curves over an interval described in th…

Ramachandran's conjecture. One of the following holds: the universal cover $\widetilde{X}$ has the Bochner–Hartogs property, or $\pi_{1}(X)$ is commensurable with the fundamental g…