32 problems

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…

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 $\mathrm{M}$ 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.

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

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

Conjecture. For every $a\in\mathbb{C}^4$, the following are equivalent: the properad $\mathcal{P}_a$ induces a confluent system; the morphism $\varphi_a$ is a bijection in weight…

Let $U$ be the superspace equipped with its nondegenerate quadratic element $\omega$, let $E$ be a vector space of dimension $n$, and set … The elements of $S^2E$ are central, and…

Let $S$ be a nonstandard graded polynomial ring, and for a positive integer $e$ let $S^{(e)}$ denote its $e$-th Veronese subring. A graded algebra is nonstandard Koszul when it has…

Koszul Roller Coaster conjecture. There exists such an algebra satisfying

Let $k$ be a flexible field and let $F/k$ be a finitely generated field extension. The isotropic Milnor K-ring of $F/k$, denoted $\operatorname{\mathrm{k}}^M_*(F/k)$, is the quotie…

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

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

Let $\mathcal{K}$ be a flag simplicial complex, and consider the grading on $H^*(\mathcal{Z}_\mathcal{K};\mathbb{Q})$ specified in the source's preceding theorem. Koszul formality…

Let $p$ be a prime and let $\mathbb{K}$ be a field containing a root of unity of order $p$. For a pro-$p$ group $G$, let $\mathfrak{L}(G)$ be the restricted Lie algebra induced by…

Let $p$ be a prime and let $\mathbb{K}$ be a field containing a root of unity of order $p$. Write $G_{\mathbb{K}}(p)$ for the maximal pro-$p$ Galois group of $\mathbb{K}$, and let…

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 $Q$.…

Let $p$ be a prime, let $\mathbb{K}$ be a field containing a root of unity of order $p$, and let $\mathcal{G}_{\mathbb{K}}$ be its maximal pro-$p$ Galois group. Suppose that…

Matroid Chow-ring conjecture. The Chow ring of any matroid is Koszul.

Wonderful-model Koszulness conjecture. The ring

Let $p$ be prime, let $F$ be a field containing a primitive $p$th root of unity, and assume that $G_F(p)$ is finitely generated. Write $H^\bullet(G_F(p),\mathbb{F}_p)$ for the cont…

Let $\mathcal B$ be a chosen block, let $P=\bigoplus_{\lambda\in\Lambda(\mathcal B)}P(\lambda)$ be a minimal projective generator of $\mathcal B$, and let … denote its locally fini…