62 problems

Let $\overline{M}_{0,n}$ be the moduli space of stable curves of genus $0$ with $n\geq 3$ marked points, let $L_1,\ldots,L_n$ be the standard cotangent line bundles, and set … With…

Let $M$ be a matroid, and let $\operatorname{CH}(M)$ denote its Chow ring. Dotsenko's conjecture. The Chow ring $\operatorname{CH}(M)$ should be a Koszul algebra. This conjecture c…

Let $p$ be a prime, let $\mathbb{K}$ be a field containing a primitive $p$th root of $1$, and let $G_\mathbb{K}=\operatorname{Gal}(\mathbb{K}_s/\mathbb{K})$ be its absolute Galois…

Uniform-matroid onset conjecture. For fixed $i eggeq 3$, the sequence $cmathcal{L}(n)i$ is representation stable past $n=2i-1$.

Let $Q$ be a symmetric quiver, and let $\mathcal{A}_Q$ be the associated supercommutative quadratic algebra. An algebra is Koszul when its ground-field trivial module admits a line…

Polishchuk–Positselski's conjecture. The algebra $A$ has at least $n$ generators.

Braid-matroid equivariant injectivity conjecture. For all $i eggeq 1$, there exist equivariant injections

Let $p$ be a prime, let $\mathbb{K}$ be a field containing a root of unity of order $p$, and, if $p=2$, suppose that $\mathbb{K}$ also contains $\sqrt{-1}$. Let $G$ denote the rele…

Let $R=Q/I$ be a Koszul algebra, and let $t_i(R)$ denote the maximal degree of a minimal generator in homological degree $i$ of the minimal graded free resolution of $R$ over $Q$.…

Weigel's Koszulity conjecture. The graded algebra $\operatorname{gr}(\mathbb F_pG_K(p))$ is Koszul. This is a companion conjecture to the expected Koszulity of the cohomology algeb…

Higher Koszul cohomology Calabi–Yau conjecture. Then $A$ is $n$-Calabi–Yau. This conjecture proposes that concentration of the higher Koszul cohomology in a single degree, with one…

Let $A$ be the algebra governing the principal block of category $\mathcal{O}$, let $\Delta$ be the direct sum of the relevant Verma modules, and let $\mathtt{E}$ be the graded ext…

Let $\Gamma$ be a connected graph with adjacency matrix $A$, and let $P({\bf \Gamma})$ be the associated preprojective algebra, constructed from a choice of field $k$ and a modulat…

Let $M_n$ denote the real locus of the moduli space of stable genus-zero curves with $n$ marked points, and let $\Lambda_n=H^*(M_n,\mathbb{Q})$. Let…

Koszul point-configuration conjecture. If

Let $C$ be a non-hyperelliptic curve of genus $g$ with a base-point free tetragonal system $T$, and suppose $C$ has the complete-intersection presentation on a three-dimensional ra…

Let $C$ be a non-hyperelliptic curve with a base-point free tetragonal system $T$, presented as a complete intersection on a three-dimensional rational normal scroll as in the prec…

Let $S$ be a finitely generated graded $\Bbbk$-algebra that is Koszul, and let $T^i_{S/\Bbbk}$ denote its $i$th cotangent cohomology, with its internal grading. The extremal-weight…

Let $\mathrm{M}$ be a matroid and let $\mathcal{G}$ be a building set such that $(\mathrm{M},\mathcal{G})$ is a flag or complete built matroid. Koszulity conjecture. The Chow ring…

Koszul filtration conjecture. The quotient $S/(G)$ has a Koszul filtration. Moreover, if $(G)$ is toric, then $S/(G)$ has a monomial Koszul filtration.

Ene–Herzog–Hibi conjecture. G-quadraticity need not imply the existence of a Koszul filtration.

Positselski--Voevodsky's conjecture. All Galois cohomology algebras $H^\bullet(G_F,\mathbb{F}_p)$ are Koszul.

Polishchuk–Positselski conjecture. The Hilbert series of $A$ is a rational function.

Positselski's conjecture. Pro-$p$ absolute Galois groups of fields containing the $p$-th roots of unity have Koszul cohomology rings.

Classification conjecture. The morphism $\varphi_a$ is an isomorphism in weight $3$ if and only if $\mathcal{P}_a$ is Koszul. Moreover, these equivalent conditions hold if and only…