16 problems

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 $(Y/T,B)$ 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.

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…

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…

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

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

Let $(X,\Delta)$ be a projective klt pair such that $K_X+\Delta$ is pseudo-effective, and let $A$ be an ample $\mathbb{Q}$-divisor on $X$. The finite generation conjecture. The adj…