38 problems

Homological stability conjecture. For every $m>1$, there is an isomorphism

Let $M_{\alpha}$ be an irreducible component of the moduli space of maps from a curve $B$ to a variety $X$, and let $\operatorname{Top}(B,X)_{\alpha}$ be the corresponding space of…

Let $X$ be an irreducible smooth complex variety. For a partition $\lambda$ of $k$, let $W_\lambda(X)$ denote the symmetric complement associated with the stratum indexed by…

Let $F$ be an infinite field, let $k$ be either $\mathbb{Q}$ or $\mathbb{Z}/l$ for a prime $l$, and assume $K_2^M(F)\otimes k=0$. Let $\mathfrak{t}_r^{(k)}(F)$ denote the homologic…

Let $F$ be an algebraically closed field. Here $\mathfrak{p}_n(F)$ denotes the primitive part of the homology group associated with $\operatorname{GL}_n(F)$, and $l$ is a prime. Th…

Let $F_n$ be the free group of rank $n$, and let … The stable rational cohomology conjecture. The stable rational cohomology of $\operatorname{Aut}(F_n)$ vanishes in every positive…

Let $R$ be a ring and let $\mathcal{IU}(R^{2n})$ denote the complex of integral unimodular vectors in $R^{2n}$, equipped with the bilinear forms specified in the source. Charney's…

Let $X$ be a smooth Fano variety over $\mathbb C$, with fixed base points on $X$ and $\mathbb P^1$. For a class $\alpha$, let $\mathrm{Mor}_*(\mathbb P^1,X,\alpha)$ be the morphism…

Alpert–Kahle–MacPherson's conjecture. The inclusion of $SF_n(R_{w,h})$ into $F_n(\mathbb{R}^2)$ induces an isomorphism

Let $G_1\to\dots\to G_n\to G_{n+1}\to\dots$ be a sequence of groups satisfying homological stability, and let $W_n$ denote the associated destabilization complexes. Converse conjec…

Let $\operatorname{Hur}_{n,b}$ denote a Hurwitz scheme parametrizing branched degree $n$ covers of $\mathbb{P}^1$ with $b$ branch points, together with the relevant monodromy and r…

Let $g\geq2$ be an integer and let $G$ be a finite group. Let $\operatorname{CHur}_g^G$ be the algebraic stack over $\mathbb C$ whose objects are geometrically connected finite éta…

Let $q$ be an odd prime power. For monic and square-free $d\in\mathbb{F}_q[t]$, let $L(s,\chi_d)$ denote the $L$-function associated to the Galois representation given by the first…

Let $M$ be an $n$-manifold, and let $\mathrm{BDiff}_c^{r}(M)^{\delta}$ denote the classifying space of the discrete group of compactly supported $C^r$ diffeomorphisms, while…

Let $\beta_n$ be the braid group, let $V_\lambda$ range over irreducible representations of the relevant symplectic group, and let a function $\theta$ be a uniform stability bound…

Let $\pi:\mathcal{X}\to B$ be a Fano fibration and let $M$ range over the Manin components of $\operatorname{Sec}(\mathcal{X}/B)$, with degree tending to infinity. Stability conjec…

Let $n\geq 3$ be odd, let $W_{g,1}$ denote the corresponding high-dimensional manifold, and consider the stabilization maps on rational homology … An optimal stability bound should…

Folklore conjecture. For each $k$, there exists some $G_k$ such that for $g \geq G_k$, the homology group $\operatorname{H}_k(\operatorname{\mathcal{I}}_g;\mathbb{Q})$ is finite-di…

Ellenberg–Venkatesh–Westerland's conjecture. For fixed $d$ and sufficiently large $g$, this map induces an isomorphism on $i$th homology. A slight variant in the cited source uses…

Let $\operatorname{GL}_n(\mathbb{Z})$ be the general linear group of invertible $n\times n$ matrices over the integers. Its stabilisation map … is induced by the inclusion…

Hyper Tor stabilization conjecture. The hyper Tor groups

For fixed $n$, let $\operatorname{Sel}_n^d(\mathbb{C})$ be the Selmer space associated with the universal family of Weierstrass models of degree parameter $d$. A Landesman homologi…

The spectral-sequence degeneration conjecture. The spectral sequence degenerates on the first page for all $m+s$ satisfying that inequality.

The stabilisation conjecture. The pairing

Let $\operatorname{Mod}_g$ be the mapping class group of a closed oriented genus $g$ surface, and for $\ell\geq 2$ let $\operatorname{Mod}_g(\ell)$ be the kernel of its action on…