21 problems

Let $Q=\mathcal L(F_\infty)$, and consider the subfactors constructed in Popa's breakthrough papers and the subfactors obtained from the non-tracial version of Popa's construction…

Let $Q$ be a factor, and consider the subfactors constructed by Popa in the cited breakthrough papers. In the case $Q=\mathcal L(F_\infty)$, these are subfactors of the free group…

Equivalence conjecture. This construction is equivalent to the construction due to A. Ocneanu.

Assume that the underlying inclusion has depth $2$, so that ${\cal M} \cap \gamma({\cal N}')$ is a factor and, by Corollary 6, … Thus it has a unique faithful normal tracial weight…

Let $X_{f,f}$ be the $M_\infty$-$M_\infty$ bimodule corresponding to the central projection $p_{f,f}/n$ in $\operatorname{Tube}{\cal M}$. Here, the dimension of a bimodule means th…

Let $N\subset M$ be a finite-index regular inclusion of type $II_1$ factors. A unitary orthonormal basis for the inclusion is a finite collection of unitaries in $M$ satisfying the…

Let $G$ be an ICC property (T) group, let $\mathcal{L}(G)$ denote its group von Neumann algebra, and let $\mathscr{I}(\mathcal{L}(G))$ be the set of Jones indices of finite-index s…

Let $N\subset M$ be a finite-index regular subfactor of type $II_1$ factors, with the associated trace-preserving conditional expectation. A unitary orthonormal basis is a Pimsner–…

The even part of the Haagerup subfactor planar algebra $H\mathcal{P}$ is a subfactor planar algebra lying in a one-parameter family of subfactor planar algebras. Haagerup family co…

Bakshi–Gupta's conjecture. Every finite-index regular inclusion of $II_1$-factors admits an orthonormal unitary Pimsner–Popa basis.

Paley type $II$ Temperley–Lieb conjecture. Paley type $II$ Hadamard matrix subfactors have Temperley–Lieb standard invariants.

Paley type $II$ principal-graph conjecture. Paley type $II$ Hadamard matrices yield subfactors with $A_\infty$ principal graphs.

Let $\mathcal C_1$ and $\mathcal C_2$ be modular tensor categories, and let $Z$ be a generalized modular invariant matrix with $Z_{00}=1$. A gapped domain wall between the correspo…

Let $\cP_\bullet$ be a unitary 2-shaded planar algebra. A finite-index homogeneous connected $\rm II_1$ multifactor inclusion is an inclusion $A\subset B$ with finite Jones index a…

Let $N\subseteq M$ be an irreducible subfactor with finite index, and let the number of minimal intermediate subfactors be counted over the inclusion $N\subseteq M$. Polynomial-gro…

Let $A_3 * A_4$ denote the corresponding Fuss–Catalan standard invariant for a hyperfinite subfactor. The higher Fuss–Catalan standard invariants are not amenable. Unclassifiabilit…

Enumeration conjecture. At index $3+\sqrt{5}$, there are exactly $13$ non-Temperley–Lieb subfactor planar algebras: the Bisch–Jones Fuss–Catalan $A_3*A_4$ subfactor planar algebra…

Morrison–Peters conjecture. There are exactly $2$ non-Temperley–Lieb subfactor planar algebras with index in $(5,3+\sqrt{5})$: the unique $\mathfrak{su}(2)_5$ subfactor planar alge…

Let a subfactor planar algebra mean the planar algebra associated with a finite-index subfactor, and let $A_\infty$ denote the infinite-depth principal graph. Consider irreducible…

Suppose that ${\cal A}\subset {\cal B}$ are conformal subnets with finite index. Let the vacuum representation of ${\cal B}$ carry the action of ${\cal A}$, and let $V$ be the spac…

Let $1$ denote the trivial representation of $SU(3)$, let $\rho$ be the fundamental representation, and let $\overline{\rho}$ be its conjugate representation. For nonnegative integ…