28 problems

Let $\Sigma_{g,n}$ be a surface of genus $g$ with $n\leq 1$ boundary components, and let $\mathcal{I}(\Sigma_{g,n})$ denote its Torelli group. Johnson's cubic-growth conjecture. Fo…

CFG conjecture. Every reductive linear algebraic group has property (CFG).

Let ${\cal B}_n^t$ be the quotient of the braid algebra ${\cal B}_n$ with parameters $t_{ij}$, and let $\mathbb{Q}[t]=\mathbb{Q}[t_{ij},\ 1\leq i<j\leq n]$. Finite-generation conje…

Let ${\cal E}_n^t$ be the quadratic algebra with central parameters $t_{ij}=[ij]^2$, and let $\mathbb{Z}[t]=\mathbb{Z}[t_{ij},\ 1\leq i<j\leq n]$. Finite-generation conjecture. The…

Let $A$ be a finite-dimensional algebra over a field $\k$, let $H^*(A,A)$ denote its Hochschild cohomology ring, and let $\mathcal{N}$ be the ideal generated by the homogeneous nil…

Let $(Y/T,B)$ be a weak log Fano contraction, in particular Klt, and let $\{\mathcal{D}_i\}_{i=1}^{\infty}$ be a system of b-divisors. Assume that $\{\mathcal{D}_i\}$ satisfies the…

Existence conjecture for log canonical models. The graded ring of $\cal O_S$-modules

Let $k$ be a field and let $\mathcal{C}$ be a tensor category of moderate growth. Write $\mathsf{Alg}\mathcal{C}$ for its category of algebra objects, and let $A^0$ denote the maxi…

For a regular semisimple $gamma$, define the graded module … It is a module over the graded algebra $oplus_{d=0}^{cinfty}{}_0\mathcal A_d$ and therefore defines a quasi-coherent sh…

Let $W(E_8)$ be the Weyl group of $E_8$, and consider all $W(E_8)$-invariant weak Jacobi forms of integral weight and integral index. Let $M_*(\operatorname{SL}_2(\mathbb Z))$ deno…

Finite-generation conjecture. The associated graded ring $\operatorname{gr}_v R$ is finitely generated.

Finite-generation conjecture. The associated graded ring $\operatorname{gr}_v R$ is finitely generated.

Twisted K3-fibration finite-generation conjecture. One has

Let … be the $((3;2),(0;0))$-twisted genus-$g$ quasimap series of ${\mathbb P}^2\times{\mathbb P}^1$, and let … with $X(q_1)=q_1(\partial B_1'/\partial q_1)/(1+B_1')$. Twisted elli…

K3-fibration finite-generation conjecture. For $\gamma\in H_2(B,\mathbb Z)$,

Let $\mathfrak{g}$ be a simple Lie algebra, and let $X$ be a stable curve in $\overline{\mathcal{M}}_{g,n}$. Write $\mathbb{V}_{X,\mathfrak{g}}^{\dagger}$ for the algebra of confor…

An arithmetic group in $\mathrm{SL}(n,\mathbb{Z})$ is a subgroup arising from the arithmetic structure of $\mathrm{SL}(n,\mathbb{Z})$. The two-generator finite-index subgroup conje…

Let $M$ be a complete noncompact $n$-dimensional Kähler manifold with nonnegative holomorphic bisectional curvature. Let $O_{P}(M)$ be the ring of all holomorphic functions on $M$…

Non-finite-generation conjecture. The group ${{\mathcal K}}_g^b$ is not finitely generated for all $g\geq 3$.

Let $H$ be a finite-dimensional Hopf algebra over a field. Its cohomology ring is … where $k$ is the trivial $H$-module. Finite generation conjecture. The graded ring…

Let $A$ be a finite dimensional Hopf algebra over a field $k$, and let $V$ be a finite dimensional $A$-module. Write $H^*(A,k)=\operatorname{Ext}^*_A(k,k)$ for the cohomology algeb…

Let $\mathcal{S}$ be a sequence of perfect 2-generated transitive permutation groups satisfying the hypotheses of Lemma 3gen: for every group $S_k$ in the sequence and every pair o…

Finite generation conjecture for log canonical rings. The log canonical ring

Let $k$ be a field, let $R=k[x_1,x_2]$ be the polynomial ring in two variables over $k$, and let $R_1,R_2$ be finitely generated $k$-subalgebras of $R$. Finite-generation conjectur…

Finite-generation conjecture. The ring $R(X;D_1,\dots,D_k)$ is finitely generated.