59 problems

The wall-crossing setup concerns a log-bounded family $\widetilde{\mathcal{G}}$ subject to a volume condition $(\clubsuit)$, under which Theorem … still holds without the volume co…

Let $(Q,W)$ be a self-dual quiver with potential, and let $\tau$ and $\widetilde{\tau}$ be self-dual weak stability conditions on the associated self-dual abelian category. Let…

Spectral-network stability-condition conjecture. The collection $\mathcal{P}$ forms a pre-slicing of the standard $\mathbb{V}_{\mathcal{S}}$-collection, and $(Z,\mathcal{P})$ is a…

Let $\mathcal{T}'$ be a $\Gamma$-labelled tree. In the Gaiotto–Moore–Neitzke asymptotic expansion, consider the total contribution to wall crossing obtained by choosing every possi…

Fix $\mathbb K=\mathbb C$. Let $P$ be a Calabi–Yau $3$-fold, set $\mathcal A=\operatorname{coh}(P)$, and let $\mathfrak F_{\mathcal A}$ and $K(\mathcal A)$ be as in the source. Let…

Let $S$ be a rational surface, let $N,d,r$ be the parameters in the wall-crossing formula, let $K^2$ denote $K_S^2$, and let $Q_i(N,d,r,K^2)$ and $P_i(N,d,r,K^2)$ be the polynomial…

Let $X$ be a smooth simply connected $4$-manifold. For a class $xi\in H^2(X,\mathbb Z)$ defining a wall, an integer $N$, and the associated wall-crossing term $\delta^X_{\xi,N}$, l…

Let $M_{Q_4}(\vec{r},n)$ be the moduli space defined using the stable condition, and let $M_{Q_4}^c(\vec{r},n)$ be the moduli space defined using co-stability, namely the condition…

Let $\mathcal A$ be the category under consideration, let $\alpha$ be a class and let $\mu$ be a stability condition, with $\varepsilon_\alpha^\mu$ the associated generalized $K$-t…

Let $n$ be a positive integer, let $\frac{r}{s}$ be a term of the $n$-th Farey sequence, and let $\tilde{M}_{r/s}$ and $\tilde{M}^{\prime}_{r/s}$ be the compositions of generalized…

Wall-crossing conjecture. There is a path $\gamma\colon[0,1]\to\Theta(Q_G,\ub{d})$ passing through walls $\mathfrak{w}_1,\dots,\mathfrak{w}_m$ satisfying the conditions in the wall…

Let $X$ be a projective threefold with Gorenstein and rational singularities, let $f:Y\to X$ be a resolution of singularities of relative dimension $1$, and let $Y$ be a projective…

Smaller-ring refinement. The preceding proposition still holds with this smaller ring $R$ in place of the ring allowing all factors $(1-Q^j)^{-1}$; in particular,

Okounkov's wall-operator conjecture. The wall operator is given by

Let $S$ be a smooth projective surface over $\mathbb{C}$, let $X=\operatorname{Tot}_S(\omega_S)$, and let $P_n^t(X,\beta)$ denote the moduli space of $\mu_t^{\dagger}$-semistable D…

Let $\mathcal{T}$ be an unrooted, unlabeled tree with $n$ vertices, and let $K_\mathcal{T}(\{\gamma_i,t_i\})$ be the tree kernel defined by the product over its oriented edges. Wri…

Let $(Q,W)$ and $(Q',W')$ be quivers with potential related by mutation at a vertex $k$, and let $d,d'$ and stability weights $\theta,\theta'$ be related by the mutation formulas i…

Wall-crossing conjecture. If $\Theta\in L_-^-(k)$ or $L_+^-(k)$ with $k\geqslant1$, then there exist choices of signs such that

Analytic stability data correspondence conjecture. The data and axioms described in 1)-8) are in bijection with the set of continuous families of analytic stability data parametriz…

Resurgence conjecture. In the canonical formal trivialization, the Taylor series of $\chi_\gamma(s(t))$ is resurgent. The same is true for the Taylor series of…

Analytic stability data conjecture. The proposition asserting exponential bounds on $|a(\gamma)|$ for $\gamma\in C(l)$ holds for any central charge $Z$, not necessarily a rational…

Let $X$ be a compact oriented $4$-manifold with $b_+^2(X)=1$, without assuming that $X$ is simply connected, and consider its Donaldson-theoretic moduli spaces and geometric data a…

Let $X$ be a Calabi–Yau $4$-fold and let $\mathcal A=\operatorname{Coh}(X)$, with the moduli, stability, orientation, homology, and virtual-class data specified in the universal en…

Let $v_0$ be a null vector and let $N_k,q_k$ be the charge data entering the pairings … Assume that all $b^{\,0}_{k\ell}$ are nonzero. Let $\Phi_n$ be the function defined in the s…

In the setting of the wall-crossing conjecture, let $P^{\mathrm{JS}}_{n,\beta}(\gamma)$ denote the Joyce–Song chamber stable-pair invariant, and let $n_{0,\beta}(\gamma)$ be the ge…