32 problems

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…

Koszul bar-to-levelled conjecture. The composition

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…

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

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…

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…

Core--cocore Koszul duality conjecture.

Arone's conjecture. The category formed by these sections is equivalent to $K(\mathsf{OEpi})$. Moreover, $\mathsf{OEpi}$ is a twisted Koszul self-dual category: $K(\mathsf{OEpi})$…

Product-rule conjecture. There is an equivalence of right $K(\mathsf{OEpi})$-modules

Symmetric monoidality conjecture. For $\mathsf{OEpi}$,

Let $\mathcal{O}$ be an operad in an ambient $\infty$-category $\mathcal{C}$, and let $B\mathcal{O}$ denote its bar construction, a cooperad. Write…

Let $\scrX$ be a hyperspherical $G$-variety and $\scrX^\vee$ its dual hyperspherical $G^\vee$-variety, with quantizations $Q\scrX$ and $Q\scrX^\vee$. Let…

Let the Higgs and Coulomb branches of a 3d $\mathcal{N}=4$ supersymmetric gauge theory be mirror dual symplectic singularities, and let $\mathcal{O}$ denote the category associated…

Let $\mathbb{X}$ and $\mathbb{Y}$ be field theories, and let $\mathsf{Obs}_{\mathbb{X}}$ and $\mathsf{Obs}_{\mathbb{Y}}$ denote their observables. For an $\mathbb{E}_n$-algebra $A$…

Let $O$ be a termwise $\Sigma$-finite, termwise cofibrant operad and let $R$ be a termwise $\Sigma$-finite, termwise cofibrant right module in $(\mathrm{Sp},\wedge)$. Write $K$ for…

Twisted holography principle. These two algebras should be Koszul dual:

Let $\mathcal{P}$ be a Hopf operad generated by primitive nilpotent elements, and let $\mathcal{Q}$ be the suboperad they generate, assumed to be Koszul. Write $\mathsf{Comm}$ for…

Let $\mathsf{FM}_2^{\mathrm{Ext}}$ be the Ext-enhanced category of free-monodromic complexes, and let $\mathrm{Tilt}_2^{\mathrm{Ext}}$ be the full subcategory consisting of…

Consider a $D$-brane in type IIA or IIB string theory, a brane in the topological string, or a brane in $M$-theory, viewed as a defect in the string theory or supergravity theory.…

Quasi-invertibility conjecture. Either or both of the DD bimodules

Let $\mathcal{O}_{\!\operatorname{g}}$ be the geometric category $\mathcal{O}$ associated with a conical symplectic resolution and a Hamiltonian torus action. Standard Koszulity co…

Let $\mathfrak{g}$ be the Lie algebra used to define the gauge theory, let $\operatorname{Obs}^q({\mathbb{C}}^\times \times {\mathbb{R}}^2)$ be the resulting quantized $E_2$-algebr…

Let $\mathsf{O}$ and $\mathsf{O}^!$ be a dual pair of binary quadratic non-Sigma operads. Let $\circ\in\mathsf{O}(2)$ be a bilinear operation and let $\circ'$ be its corresponding…