48 problems

Let $H(\theta)$ be the one-parameter family of self-adjoint complex Hadamard matrices of order $6$ defined above, with…

Let $\Gamma_k$ denote the graphs in the paper's Figure 2. Haagerup graph principal-graph conjecture. For every $k>1$, the graph $\Gamma_k$ is not a principal graph of a subfactor.…

Let the graphs in the paper's Figure 2 be indexed by $k$, with the parameter restricted to $k>1$. Haagerup graph conjecture. For every $k>1$, neither graph in the relevant pair is…

The paper considers Haagerup's candidate graphs, with the parameter $n$ indexing the graphs in the family described in the paper. Haagerup graph obstruction conjecture. None of the…

Let $L(7,1)$ and $L(7,2)$ be the indicated lens spaces, and suppose there exists a generalized $E_6$-subfactor with group symmetry $G=\mathbb{Z}/7\mathbb{Z}$. Let the Turaev–Viro–O…

Let $X$ be an irreducible, hyperfinite subfactor of finite depth, and let its Jones index be $[X]$. A subfactor has Temperley–Lieb–Jones (TLJ) standard invariant when its standard…

Let $N\subset M$ be a regular subfactor, meaning that the normalizer $\mathcal{N}_M(N)$ generates the von Neumann algebra $M$. A unitary orthonormal basis for $N\subset M$ is a uni…

Let $N \subset M$ be an inclusion of von Neumann algebras with expectations. For a strictly semifinite weight $\varphi \in \mathcal{P}(N)$, let…

Let $N \subset M$ be an irreducible inclusion of factors with expectation and with separable preduals, meaning that $N' \cap M=\mathbb{C}$ and there is a faithful normal conditiona…

Let $G$ be a sharply $k$-transitive permutation group, meaning that $G_{x_1,x_2,\dots,x_{k-1}}$ acts regularly on $X\setminus\{x_1,x_2,\ldots,x_{k-1}\}$. Goldman-type theorem for s…

Ufc-amenability conjecture. Any hyperfinite $A_\infty$-subfactor $N\subset M$ is ufc-amenable, and consequently

Let $\mathcal{C}$ be a unitary fusion category; equivalently, let it arise from a finite-depth subfactor. Let $\mathcal{A}$ be a completely rational conformal net, regarded as a re…

Let $N\subset M$ be a finite-index regular inclusion of factors of type $II_1$. A unitary orthonormal basis conjecture asserts that $M/N$ has a unitary orthonormal basis. The quest…

Let $\mathcal{N}\subset\mathcal{M}$ be a discrete subfactor, let $\phi\in K(\mathcal{N}\subset\mathcal{M})$, and let $b_\phi$ and $\dim(\phi)$ be the scalar and hyperdimension asso…

Let $\mathcal{N}\subset\mathcal{M}$ be a discrete subfactor and let $K(\mathcal{N}\subset\mathcal{M})$ be its associated compact hypergroup. For…

Let $K$ be a compact hypergroup acting faithfully and minimally on a von Neumann algebra $\mathcal{M}$, with fixed-point algebra $\mathcal{M}^K$. The inclusion…

Let $\b5$ be the parameter of the one-parameter family of three-dimensional canonical fusion bialgebras in case II, with $\b5\rfloor 3$. A fusion bialgebra is subfactorizable if it…

Let $M=L(\mathbb F_t)$ be an interpolated free group factor with $1<t\leq\infty$. A hyperfinite subfactor $L\subset M$ is called ergodic when the inclusion has the ergodicity prope…

Strictly outer action conjecture. Any strictly outer action $G\curvearrowright Q$ on a factor of any type satisfies the Intermediate Subfactor Property.

A strongly amenable graded hyperfinite subfactor of type II$_1$ is a subfactor whose grading, amenability, hyperfiniteness, and type are as specified; a subfactor planar para algeb…

$AH+2$ Q-system conjecture. The construction of $AH+1$ can be iterated once more: there is a Q-system for

Quadrilateral conjecture. There should be another quadrilateral whose upper inclusions are each $AH+1$ and whose lower inclusions have index one larger.

Let $D(E_6)$ denote the quantum double, or Drinfeld center, of the $E_6$ subfactor. Let … be the indicated non-simply connected gauge group. Chern--Simons relation conjecture. The…

Let $\mathcal{F}$ be a unitary fusion category, and let $Z(\mathcal{F})$ denote its Drinfeld center. For a finite-depth subfactor $N\subset M$, let $D(N\subset M)$ denote its quant…

A finite-depth subfactor has index $3+\sqrt{5}$ if its Jones index is equal to $3+\sqrt{5}$. Up to duality, two additional finite-depth subfactors with this index are known. Nonexi…