17 problems

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 $L(7,1)$ 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 $N \subset M$ 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 inclusion of von Neumann algebras with expectation, and let $\varphi \in N_*$ be a faithful normal state. Write $c(N)=N \rtimes_{\sigma^\varphi} \mathbb{R}$…

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…

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 $\b5$ 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…

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

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…

Morrison–Peters conjecture. Any subfactor with index in the range $(5,3+\sqrt{5})$ has principal graphs $(A_\infty,A_\infty)$, the pair of graphs represented in the source by…

Classification conjecture. Any subfactor with index in $(5,3+\sqrt{5})$ has principal graph $A_\infty$, $\Gamma(\mathcal A)$, or $\Gamma(\mathcal B)$.

Let $N\subset M$ be a finite-depth subfactor, let $P$ be its associated subfactor planar algebra, and let the $N$-$N$-bimodule category generated by $_N L^2(M)_M$ be the category w…

Algebra-object decomposition conjecture. The algebra object satisfies

Tensor-product conjecture. The number of minimal intermediate subfactors in $N_1\otimes N_2\subset M_1\otimes M_2$ not of either excluded form is at most

Fusion-algebra version of Wall's conjecture. The number of maximal fusion subalgebras generated by a subset of the simple objects of ${\cal F}$ is less than $n$.

Linear-independence conjecture. There exist vectors $\xi_i,\xi\in N'\cap M_1$ such that