13 problems

Let $V$ be a reflection-faithful representation of ${\cal W}$ over an infinite field, and let $R$ be the ring of regular functions on $V$. For $x\in{\cal W}$, let $C'_x$ be the cor…

Let $\lModl$ be the 2-category of loosely discrete opfibrations and $\TwModl$ the 2-category of twisted bimodules, with the 2-functor $\twsection{-}: \lModl \to \TwModl$ given by t…

Let $(\Theta,\Phi)$ be a tangential pair, let $F$ be the fibre of $\Theta \to \Phi$, let $A$ be the Thom ring spectrum of the index bundle over $\Omega F$, and let $R$ be the Thom…

Let $\Theta$ be a tangential structure and let $R$ be the ring spectrum defined from it. Let $\operatorname{Flow}^{\Theta}$ denote the category of $\Theta$-oriented flow categories…

Let $D$ be the algebra of dual numbers, let $\mathscr{D}$ be the associated 2-category of projective bimodules, and for each positive integer $k$ let $\mathcal{J}_k$ denote the two…

The paper constructs bimodules $\mathcal{X}^{DA}$, $\mathcal{X}^{DD}$, and $\mathcal{X}_i^{DD}$, together with the Heegaard diagram in Figure and its variant globally extended to t…

Let $A$ be an algebra and let $M$ be an invertible $A$-bimodule, with inverse $M^{-1}$. Write $4\widehat{A}4$, $4\widehat{M}4$, and $4\widehat{M^{-1}}4$ for their Arens–Michael env…

Let $\mathcal{C}$ be a unitary fusion category, and let $R$ be a hyperfinite factor which is either of type ${\rm II}_1$ or ${\rm III}_1$. A fully faithful representation of…

Bimodule generation conjecture. The $D(H)$-bimodule $\mathcal{H}_{\lambda,\mu}$ is generated by $h_{\lambda,\mu}$. The conjecture is based on the examples computed in the paper, an…

Let $t\in(1,\infty]$ and let $N\subseteq L(\mathbb{F}_{t})$ be a maximal amenable von Neumann subalgebra. Regard the following as $N$-$N$ bimodules, where…

Let $\mathcal{C}$ be a nice $\use@mathgroup \M@U \symAMSb{E}_{n}$-monoidal $\infty$-category. Let $\mathfrak{Alg}_{n}^{\textup{FA}}(\mathcal{C})$ denote Scheimbauer’s…

Let $C$ be a tilted algebra, and let $E=\operatorname{Ext}_{C}^{2}(DC,C)$. Regard $E$ as a $C$-$C$-bimodule and decompose it into indecomposable summands. Orthogonality conjecture.…

Let $A$ and $B$ be algebras, and let $X$ and $Y$ be $A$-$B$-bimodules. A bimodule homomorphism $f:X\to Y$ determines the 2-diagram described by the maps involving $Z(A)$, $Z(B)$, a…