112 problems

Let $\mathcal C$ be a fusion category over an algebraically closed field of characteristic zero. A $Spin(3)$ homotopy fixed point structure on $\mathcal C$ is the additional struct…

Tensor-product equivalence conjecture. The map above induces an equivalence of monoidal categories

Let $V$ be a finitely strongly generated, self-dual, quasi-lisse vertex operator algebra of CFT type, and let $\mathcal{C}_V^{\mathrm{ord}}$ denote its category of ordinary modules…

Benson's conjecture. All indecomposable summands of $V\otimes V^*$ have dimension divisible by $2$, except for the single summand isomorphic to the trivial representation $\Bbbk$;…

Let $H$ be a semisimple Hopf algebra over $\mathbb{C}$. Kashina's conjecture. The exponent of $H$ divides $\operatorname{dim}(H)$. This conjecture is a divisibility analogue of the…

Conjecture on $L_{k_m}(D_4)$. For each $m\geq1$, $L_{k_m}(D_4)$ is a quasi-lisse vertex algebra, it has a unique irreducible ordinary module, and $KL_{k_m}(D_4)$ is a semisimple br…

Let $H$ be a Hopf algebra, let $B$ be a left coideal subalgebra, and let $(H/B^+H)^\ast\#B$ be the associated left partially dualized quasi-Hopf algebra. In Remark on invertibility…

Braided tensor equivalence conjecture. The functor $H_f^0$ is an equivalence of braided tensor categories. The source already states that this functor is an equivalence of abelian…

Grothendieck-ring growth-dimension conjecture. The function

G-crossed tensor-category conjecture. This category has a natural structure of a $G$-crossed tensor category satisfying additional properties.

Let $\mathcal C$ be a pivotal fusion category over a field of characteristic zero. An $SO(2)$ homotopy fixed point structure on $\mathcal C$ is the additional structure correspondi…

Let ${\cal SW}(V)$ be the Schur–Weyl category, let $V_{\lambda}$ denote its distinguished objects, and let ${\bf g}=sl(V_S)$ act on these objects. A linear map between two objects…

Tensor-functor realization conjecture. For every prime $p$ there exists a tensor functor

The modularity conjecture. The subcategory ${\mathcal M}_e$ is closed under convolution and hence is a monoidal category with unit $e$ if the preceding perversity conjecture holds;…

Let ${\mathcal C}_\zeta^{{\underline c}_f}$ be the monoidal category associated with a finite two-sided cell ${\underline c}_f\subset W_f(\zeta)$, and let…

Rigidity conjecture. (i) Any fusion ring has at most finitely many realizations over ${\bf k}$, up to equivalence (possibly none). (ii) The number of tensor functors between two fi…

Let $Sp^{\Sigma}$ be the category of symmetric spectra and let $Comm(Sp^{\Sigma})$ be the model category of commutative ring spectra. Let $LSp^{\Sigma}$ be the tensor Segal categor…

Let $Sp^{\Sigma}$ be the category of symmetric spectra with its positive model structure, and let $Comm(Sp^{\Sigma})$ be the model category of commutative ring spectra. Write…

Let ${\mathcal C}$ be a finite tensor category, let $V\in{\mathcal C}$, let $V^{**}$ denote the double dual of $V$, and let $L_\rho$ be the distinguished invertible object of…

Circulator trace conjecture. In the case of stable Gelfand–Kazhdan categories,

Let $\mathcal{C}$ be a finitely tensor-generated pre-Tannakian category of moderate growth. Write $\mathcal{M}\mathcal{Z}(\mathcal{C})$ for its Müger center, and let…

Let $\mathfrak g$ be a semisimple Lie algebra with weight and root lattices $P$ and $Q$, respectively. For $q\in\mathbb{C}$, let $U_q(\mathfrak g)\mathbf{-mod}$ be the category of…

Crystal Temperley–Lieb center conjecture. The canonical functor $G$ is a monoidal equivalence. This conjecture asserts that the only objects in the Drinfeld center are those induce…

Let $k$ be a field, let $\mathbf Q(\mathbb Z^3)$ be the QCA space in dimension three, and let $\mathrm{Witt}_k$ denote the Witt group of braided fusion $k$-linear categories. The Q…

Let $\mathfrak{g}$ be a Lie algebra, let $e\in\mathfrak{g}$ be distinguished nilpotent, and let $Q=Q(e)$ be the finite centralizer of an $sl_2$-triple associated with $e$ in the si…