19 problems

Let $\mathbb{T}^n_\theta$ be the quantum torus, let $\Delta$ be its flat Laplacian, let $q>0$, set $r=2nq^{-1}$, and suppose that either $r\neq 1$ and $r'=\max(r,1)$, or $r=1<r'$.…

Let ${\cal C}$ be an $A_{\infty}$-category such that $HH^1({\cal C})$ is abelian and the connected automorphism group $G_{\cal C}=Aut^0({\cal C})$ is a finite-dimensional commutati…

Deformation conjecture. The simple module $Z_\chi$ can be deformed, for each $v\neq 0$, to a simple object of $\mathcal{M}_{q,t}(\mathsf{H}_v,\mathbb{C}\boldsymbol{X})$.

Let $\widetilde{\psi}\colon\mathcal{D}'_{\mathfrak{sl}_3}\to A_3(\mathbb{C}_{\epsilon}(t))[\mathbf{x}_1^{-1},\mathbf{x}_2^{-1},\mathbf{x}_3^{-1}]$ be the algebra homomorphism defin…

Let $L$ be a link, let $\mathsf{H}_q(L)$ be its HOMFLYPT difference module, let $[L]$ be its distinguished link class, and let $\mathsf{ev}\colon \mathsf{H}(L)\to \mathcal{F}_r$ be…

Let $L$ be a link, let $\mathsf{H}_q(L)$ be its HOMFLYPT difference module, let $\mathsf{ev}\colon \mathsf{H}_q(L)\to \mathcal{F}_r$ be the evaluation map to colored HOMFLYPT polyn…

Let $C\in M_{\mathbb{I}\times\mathbb{I}}(\mathbb{Z})$ be a generalized Cartan matrix of finite type, let $W(C)$ be its Weyl group, and let $\mathbf{i}=(i_1i_2\cdots i_r)$ be a word…

Let $Q$ be a quiver, let $Q_\mathbb{Z}$ be its integral level quiver, and let $\mu(\mathbf{e})=\Theta(\mathbf{e})/\dim \mathbf{e}$ denote the slope of a dimension vector…

Norm-equivalence conjecture. For $1\leq p,q\leq\infty$, this Besov quasi-norm is equivalent to the norm in Definition 3.1 of XXY.

The relevant standard tori of an easy compact quantum group $G$ are considered in the generalized standard torus setting. The Fourier tori are the standard tori associated with the…

Let $\mathcal{I}(q)$ be the Schur index obtained from the regulated expression … and let $\mathcal{I}_N(q)$ be the truncated index obtained by setting $X_{a_i\gamma_i}$ to zero whe…

Let $L$ be a UV line defect, let $\vartheta$ be a central-charge phase, and let $F(L,\vartheta)=\sum_{\gamma\in\Gamma}\overline{\underline{\Omega}}(L,\vartheta,\gamma,q)X_\gamma$ b…

Let $U$ be the quantum group, $Z$ its center, $U_0$ its Cartan subalgebra, $F_l(U)$ the relevant left fraction algebra, and let $\mathcal O_q[G/H]$ and $T$ be the quantum homogeneo…

Endpoint complex interpolation conjecture.

Let $F$ be a field, let $A$ be a finitely generated free abelian group, and let $F \operatorname{\ast} A$ be the twisted group algebra underlying the quantum torus $P(\Lambda)$. Le…

Kontsevich–Soibelman integration-map conjecture. There exists an $L$-graded $\Lambda$-algebra homomorphism

Let $V$ be a convex sector of angle smaller than $\pi$ in the central charge plane, and let $A_V^{\rm mot}$ be its motivic wall-crossing operator. Conjugation by this operator defi…

Absence of poles conjecture. For any $3d$ Calabi–Yau category with a stability condition and any strict sector $V$, the automorphism $x\mapsto A_{V,q}xA_{V,q}^{-1}$ preserves the s…

Let ${}^{\bullet}\wp^{\mathrm{nc}}_{\mu,\theta}$ and ${}^{\bullet}\wp'_{\mu,\theta}$ be the quantum nonstandard Weierstrass function and its derivative, and let…