59 problems

Let $Q$ be a quiver, let $\overline Q$ be its doubled quiver, and write … where $H^{\mathrm{BM}}$ denotes Borel–Moore homology. Let $Y_{MO}^{Q,+}$ denote the positive half of the M…

Let $\bar{\rho}=\bar{\rho}_{\theta,\theta'}$ be the sign-twist-independent map denoted by $\Phi$, from the negative half $\mathbb{Y}_\mathcal{Q}^-$ of the Yangian to the equivarian…

Let $X$ be a smooth surface and let $Z_1,Z_2\subset X$ be possibly reducible curves such that $Z_1\cap Z_2$ is zero-dimensional. Write $\mathbf{HA}^T_{X,Z}$ for the $T$-equivariant…

Let $Q$ be a quiver with no loops or $2$-cycles, and let $W$ be an infinitely mutable potential on $Q$. Write $\operatorname{\mathcal{H}}_{Q,W}$ for the cohomological Hall algebra…

Let $Q$ be a quiver, let $\mathbb{Y}_+$ be the positive half of the Maulik–Okounkov Yangian, let $K$ be the scalar extension used in the paper, and set…

Let $Q$ be a finite symmetric quiver with vertex set $I$, and let $({\mathcal H},\star)$ be its $\mathbb{Z}_{\geq 0}^I\times\mathbb{Z}$-graded super-commutative Cohomological Hall…

Nekrasov's localized wheel-condition conjecture. The inclusion of the tripled-quiver CoHA image into the wheel-condition algebra becomes an equality after localization:

Nekrasov's wheel-condition generation conjecture. The localized wheel-condition algebra equals the localized spherical subalgebra:

Let $Z=Z_1\cup\cdots\cup Z_s$ be a union of curves, let $\langle Z\rangle$ be the relevant semigroup of classes, and let $\overline{\mathfrak a}_\gamma$ be the amalgamation map in…

Let $S$ be the surface in the preceding setup, let $C_i$ be one of the distinguished curves, and let $[\mathfrak Y_{i,(d)}]$ and $[\mathfrak Z_{i,\ell}]$ denote the corresponding d…

Let $X$ be a smooth surface, let $Z_1$ and $Z_2$ be curves with $Z=Z_1\cup Z_2$, and let $\mathbf{HA}^T_{X,Z}$ denote the corresponding $T$-equivariant cohomological Hall algebra.…

Let $Q$ be a quiver and let $W$ range over quasi-homogeneous infinitely mutable potentials. Write $\operatorname{\mathcal{H}}_{Q,W}$ for the associated cohomological Hall algebra.…

Nilpotent independence conjecture. The nilpotent CoHA $\operatorname{\mathcal{H}}^{\operatorname{nilp}}_{Q,W}$ should be independent of $W$. This is proposed as a way to exclude th…

Let $Q$ be the quiver with potential considered in the paper, let $\operatorname{\mathcal{H}}_{Q,W}$ be its cohomological Hall algebra, and let $\mathcal{S}_Q$ be the spherical sub…

Let $(\mathcal{E},\otimes)$ be a $\mathcal{G}$-monoidal dg category with t-structure, for $\mathcal{G}$ as in the source. This induces an $\mathbb{E}_1$-factorisation cosheaf of ca…

BPS complex conjectures. The following assertions should hold:

Let $K>2$, and let $Y^{K}$ be the $C[Chbar_1,Chbar_2]$-linear algebra obtained by gluing the two copies of the double deformed current algebra $A^K$ constructed from observables on…

Let $Q$ be a quiver, let $A$ be its CoHA, and let $\vec f,\vec f'$ be framing vectors. Denote by $\mathrm{sH}(Q;\vec f)$ the corresponding spherical CoHA bimodule, by…

Let $A_u$ be an algebra of BPS states for a SQFT and let $B_{L;u}$ be the associated collection of $A_u$-bimodules for a line defect $L$. Let the map $\operatorname{RG}_u$ assign t…

Let $X$ be a two-sided Hecke pattern of rank $r$. Write $H_*(X)$ for its Borel–Moore homology, and let $W_{\star,\star}(S)$ denote the algebra with the appropriate choices of indic…

Shuffle-algebra image conjecture.

Generic-torus conjecture. Theorems A and B of the cited reference should also hold under this weaker condition.

Let $r,d\in\mathbf{Z}$ with $r>0$, and let $\mathscr{A}_{\mathbf{Z}_{\geqslant1}\cdot(r,d)}^{\operatorname{Betti}}$ be the indicated Betti CoHA subalgebra object. Let…

Let $r,d,d'\in\mathbf{Z}$ with $r\geqslant1$. For positive multiples of $(r,d)$, write … and let the corresponding algebra object be obtained by the symmetric product with the equi…

Let $f\colon\mathcal{C}\to S$ be a family of smooth projective curves of genus $g$ over a smooth connected base $S$. Let $\mathfrak{M}_{r,d}^{\operatorname{Dol}}(\mathcal{C})$ be t…