27 problems

Let $\mathcal{C}$ be a hereditary finitary category. A derived functor is called friendly when it sends every object into two adjacent cohomological degrees. Automorphism conjectur…

Calabi–Yau Drinfeld-double conjecture. The assignment

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories. An exact functor $F:D^b(\mathcal{C}_1)\to D^b(\mathcal{C}_2)$ is friendly…

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories. Their reduced Drinfeld doubles are denoted by…

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories, and let $\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}_i}$ denote their r…

Positivity conjecture. For any object $M$ and any exceptional object $K$ of ${\mathcal C}$, the integer $r(M,K)$ is nonnegative.

Let $n$ be the fixed positive integer in the construction, let $\mathbf{K}\simeq \Gamma\otimes\mathbb{C}[v,v^{-1}]$, and let $\langle\;,\;\rangle_G$ denote Green's scalar product.…

Let $\mathbf{U}^+$ be the $\mathbb{C}_v$-subalgebra of $\mathbf{U}_v(L\mathfrak{gl}_p)$ generated by $\mathbf{U}_0^+$ and $h_{0,l}$ for $l>0$, where $\mathbf{U}_0^+$ is generated b…

Positivity conjecture. For every $b\in\mathbf{C}^\imath$, all coefficients $f_c(v)$, $g_{b',c}(v)$ and $h_{c,c'}(v)$ belong to $\mathbb{N}[v^{\frac12},v^{-\frac12}]$. In particular…

Let $Q$ be a quiver, let … is a free $R$-module. The conjecture is used to obtain the integral-form isomorphism with the positive half of the Maulik–Okounkov quantum loop group; th…

Let $n\in\mathbb{N}$, let $(m,n)\in\mathbb{Z}\times\mathbb{N}$ be coprime, and let $d\geq 1$. For the equivariant derived category…

Subquotient conjecture. The explicit embedding for quantum groups in Theorem is a section of a subquotient. This conjecture is motivated by the corresponding result for Hall algebr…

Let $Q$ be a quiver and let $\mathbf{d}\in\mathbf{N}^{Q_0}$ be a dimension vector. The polynomial $C^{\mathrm{abs}}_{Q,\mathbf{d}}(q)$ is the absolutely cuspidal polynomial associa…

Let $Q$ be an acyclic quiver and let $\mathbf{d}\in\mathbb{Z}_{\geq 0}^{Q_0}$. For each $\mathbf{e}\leq\mathbf{d}$, let $[Q,\mathbf{e}]$ denote the relevant commuting-pair count in…

Let $g\geq 2$, let $\Sigma_g$ be a genus-$g$ surface, and let $p\colon \mathfrak{M}_{g,r}^{\operatorname{Betti}}\to \mathcal{M}_{g,r}^{\operatorname{Betti}}$ be the semisimplificat…

Let $\Lambda$ be an arbitrary $\tau$-tilting finite algebra. Let $G(\Lambda)$ be the group associated with the relations from 2-simple minded collections, and let…

Let $(Q,\tau)$ be an acyclic $\imath$quiver, let $W_{\tau}$ be the restricted Weyl group, and let ${\rm Br}(W_{\tau})$ be the braid group generated by elements $t_i$ for…

Let $(S,M,A)$ be a graded marked surface with a full arc system $A$. Say that it has enough marked intervals when … and, for each $k$, the inclusion $\iota_k:(D^2_k,M_k)\to(S,M)$ i…

Let $(S,M)$ be a graded marked surface with a full arc system $A$, and suppose that $(S,M)$ has enough marked intervals, meaning that for each disk $(D_k^2,M_k)$ in the decompositi…

Let $\mathcal{H}^{\mathrm{sph},\geq 0}_{\nu}$ be the subspace of the spherical Hall algebra component $\mathcal{H}^{\mathrm{sph}}_{\nu}$ consisting of functions supported on…

Let $A_{g,r,d}(0)$ be the constant term of the polynomial counting absolutely indecomposable vector bundles of rank $r$ and degree $d$, and let $\mathfrak{h}^{\mathrm{sph}}$ be the…

Let $Q$ be a simply laced quiver, let $\mathcal{A}={\rm Rep}(Q)$ be the category of finite-dimensional representations over a finite field $\mathbb{F}_q$, and let $\mathcal{A}_0$ b…

Let $\mathscr A$ be a profinitary exact category, and let $\Gamma_{\mathscr A}$ be its Grothendieck monoid. A monoid is locally finite if, for every $\gamma\in\Gamma_{\mathscr A}$,…

Let $k=\mathbb{C}$, let $X$ be a smooth projective curve of genus $g$, let $T^a_g$ be the torus associated with the Frobenius-eigenvalue parameter space, and let…

Let $X$ and $X'$ be generic smooth projective curves of the same genus $g$, and let $\mathcal{K}_X$ and $\mathcal{K}_{X'}$ be their graded Grothendieck algebras over…