74 problems

Let $\mathfrak{M}$ be the Higgs branch of a three-dimensional ${\cal N}=4$ supersymmetric Yang–Mills theory, and let $\mathfrak{M}^{!}$ denote its Koszul dual when this dual exists…

Symplectic duality conjecture. There exist many pairs of symplectic resolutions for which various dualities hold, including Koszul duality between their categories $\mathcal O$.

Francis–Gaitsgory's conjecture. To obtain an equivalence of $\infty$-categories, it should suffice to restrict $B_{\mathcal{O}}$ to pro-nilpotent $\mathcal{O}$-algebras and…

Quadratic Koszul reconstruction conjecture. If the relations of $A$ are purely quadratic, then dualising the Yoneda product should give a presentation of $A$.

Aguiar–Loday's conjectures. The first conjecture concerns the dimensions of the $\mathbb{S}_n$-modules $\mathcal{Q}uad(n)$, and the second conjecture is that the operad…

Let $V$ be a $K$-vector space, and let $m\text{-Tetra.}(V)$ be the free $m$-tetrahedral algebra over $V$. Write $H\overset{m}{\treeg}_n(m\text{-Tetra.}(V))$ for its homology. The f…

For a $K$-vector space $V$, let $\bigtriangleup(V)$ be the free triangular algebra over $V$, and let $H\treeg_n(\bigtriangleup(V))$ denote its homology. The first homology is…

Let ${\cal P}$ be a quadratic Koszul operad. Let ${\cal S}^*_{{\cal P}_\infty}$ be the suboperad of natural operations generated by the restricted class of operations generalizing…

Let ${\cal P}$ be a quadratic Koszul operad, and let ${\cal B}^*_{{\cal P}_\infty}$ be the differential graded operad of natural operations on the complex defining the operadic coh…

Let $P_+$ denote the set of dominant weights, let $V_\lambda$ be the corresponding module, and let $S_\sigma(V_\lambda)$ and $\Lambda_\sigma(V_\lambda^*)$ be its braided symmetric…

Koszul bar-to-levelled conjecture. The composition

Bar-to-levelled construction conjecture. The morphism $e$ is a quasi-isomorphism.

Let $\boldsymbol\Lambda$ and $\mathbf S$ be the Koszul-dual algebra and coalgebra appearing above, and let weak $(\boldsymbol\Lambda,\mathbf S)$-modules and weak $\mathbf S$-comodu…

Let an $L$-molecule be an entanglement of several codipterous coalgebras and/or anti-codipterous coalgebras constructed from coassociative coalgebras via channel maps, and let an o…

The Temperley–Lieb algebra acts on $V_1^{\otimes n}$, and the paper gives two categorifications of this action: one by Zuckerman functors in singular blocks and one by projective f…

Let $\sT\subset B$ be a Cartan subgroup, let $\lambda$ be a regular coweight of $G$, and let $\sT_{\lambda}$ be the subgroup of $\sT$ generated by $\exp(2\pi i\lambda)$. Let…

Let $B\subset G$ and $B^{\vee}\subset G^{\vee}$ be Borel subgroups, and let $D^{\mathbb Z/2}$ denote the $\mathbb Z/2$-graded version of a graded category. $\mathbb Z/2$-graded Kos…

Let $G$ be a connected reductive group, let $\fY$ be the spherical variety above, and let $B\subset G$ and $B^{\vee}\subset G^{\vee}$ be Borel subgroups. Let…

Let $\mathscr{P}$ be an operad, let $\mathbf{D}(\Bbbk)$ be the derived category over a field $\Bbbk$, and consider the enhanced bar-cobar adjunction … Here the left-hand category c…

Koszul-dual duality conjecture. The derived linear dual of the dg $\operatorname{CE}_\bullet(\operatorname{\mathfrak g})$-comodule…

Let $\operatorname{\textsl{Yam}}$ be the Yamaguti operad, presented by defining relations that are combinations of compositions of at most two generators. Quadraticity conjecture f…

Let $G$ be a semisimple group with Langlands dual group $G^\vee$, let $\mathfrak{u}_q$ be the small quantum group of $G^\vee$, and let…

The small quantum group and the semi-infinite flag variety are related to affine Springer fibers through the proposed geometric and categorical constructions described in the paper…

Let ${\mathcal A}$ be the algebra of local operators in a perturbative 3d holomorphic-topological QFT, and let ${\mathcal A}^!$ be its Koszul-dual $A_\infty$ algebra. A dg-shifted…

The holomorphic-topological Poisson sigma model has transverse Dirichlet and Neumann boundary conditions. The Neumann boundary algebra quantizes the starting Poisson vertex algebra…