15 problems

Let $R$ be a Gorenstein singularity, let $M$ be a maximal Cohen–Macaulay $R$-module, and set $A:=\operatorname{End}_R(R\oplus M)$. Let $e\in A$ be the idempotent corresponding to…

Realization conjecture. There exist an exact category $\mathcal{B}$ of finite homological dimension and a group $H$ acting on $\mathsf{D}^{b}_{\mathrm{dg}}(\mathcal{B})$ such that…

Benson's nucleus-support conjecture. The nucleus of $G$ is the same as the support of the singularity category $\mathcal{D}_\text{sg}(C^*(BG))$.

Let $\Lambda$ be an artin algebra, and let $\mathbf{D}_{\rm sg}(\Lambda)$ denote its singularity category. In a triangulated category $\mathcal{T}$, an object $M$ is presilting if…

Let $R$ be a commutative noetherian ring. Denote by $\operatorname{D_{sg}}(R)$ the singularity category of $R$, defined as the Verdier quotient of the bounded derived category of f…

Nucleus-support conjecture. The support of

Let $\mathcal{T}$ be a triangulated category, and let a presilting subcategory be a full additive subcategory $\mathcal{M}$ such that ${\rm Hom}_{\mathcal{T}}(M, \Sigma^n(M))=0$ fo…

Let $V$ be a smooth K3 or abelian surface and let $Z\subset V$ be an at-worst-nodal ample divisor. Set $U=V\setminus Z$. Nodal-divisor mirror conjecture. The category…

Let $X$ be a proper normal crossings surface with graph-like singular locus described by a trivalent graph $G(X)$ and orientable dual intersection complex. For each edge $e$ whose…

Let $X$ be a normal crossings surface with graph-like singular locus and orientable dual intersection complex, arising as the zero locus of a section $s$ of a line bundle $L$ on a…

Vanishing conjecture. The Grothendieck group of the singularity category vanishes:

Let $A$ be an algebra with idempotent $e$, let $M$ be the corresponding maximal Cohen–Macaulay module, and write … for the derived quotient. Assume that $A/AeA$ is an Artinian loca…

With notation as above, let $R$ be a commutative ring, let $A$ be the algebra occurring in the construction of the singularity functor, and let $M$ be the corresponding maximal Coh…

Let $G < \operatorname{GL}(2, \mathbb{C})$ be a nonabelian finite subgroup and let $R=\mathbb{C}\llbracket x,y\rrbracket^G$. A finite-dimensional local $\mathbb{C}$-algebra is an a…

Consider supermanifolds, their Hodge numbers, and the associated categories of singularities. A gap is a missing interval in the generation-time spectrum of such a category, and a…