12 problems

Cerulli–Labardini–Schröer conjecture. Every component cluster has cardinality at most $n$, and the $E_A$-rigid component clusters are precisely the component clusters of cardinalit…

Let $X:f=0$ be a hypersurface in $\mathbb P^n$ with Jacobian algebra $M(f)$, and let the graded Betti numbers of a minimal resolution of $M(f)$ satisfy the relations in equations a…

Asai–Iyama's conjecture. If $(Q,W)$ is a non-degenerate Jacobi-finite quiver with potential, then

Let $C(W_n^I)$ be the potential associated with the cyclic quiver construction, for parameters $m,n\in\mathbb{Z}_n$. Non-degeneracy conjecture. For any $m,n\in\mathbb{Z}_n$, the po…

Let $S=\mathbb{C}[x_0,\ldots,x_n]$ be the graded polynomial ring in $n+1$ variables, with $n\geq 2$, and let $f\in S_d$ be a homogeneous polynomial such that the hypersurface…

Let $(Q,\mathcal{P})$ be a quiver with nondegenerate potential such that its Jacobian algebra $A$ is finite-dimensional and cluster-finite, and let $M$ be the direct sum of all…

Let $A$ be a finite-dimensional Jacobian algebra. Consider the three sets of real Schur representations: all real Schur representations; those constructed from the stated Schur-rep…

Let $A$ be a finite-dimensional Jacobian algebra, let $M=\operatorname{Coker}(\delta)$, and let ${{\sf V}}(M)$ denote the set of vertices of the Newton polytope of $M$. A dimension…

Let $\mathbb{I}_n$ be the algebra of polynomial integro-differential operators and let $\mathbb{A}_n$ be the corresponding Jacobian algebra. Write ${\rm gldim}(R)$ for the common l…

Let $f$ be an invertible polynomial and let $G$ be a commutative group acting as in the preceding definitions. Write ${\rm Jac}'(f,G)$ for a $G$-twisted Jacobian algebra of $f$, an…

Cerulli–Labardini–Schröer conjecture. The indecomposable strongly reduced components of Jacobian algebras parametrize a basis of the corresponding Caldero–Chapoton algebra.

A quiver with a polynomial potential determines a Jacobian algebra via the cyclic derivatives of the potential. For a Dynkin quiver with a discrete stability structure, Keller's qu…