20 problems

Let $Y_1$ be a cobordism from $\Sigma_1$ to $\Sigma_2$ and $Y_2$ a cobordism from $\Sigma_2$ to $\Sigma_3$, with adapted harmonic Morse functions, compatible…

Quantum Kirwan map conjecture. There exists a map

Let $\Sigma$ be a surface and let $L,L'\subset\Sigma$ be disjoint circles. Write $\widehat{V}_L^n$ and $\widehat{V}_{L'}^n$ for the associated Lagrangian correspondences between th…

Let $\mathbb L_H(d)$ be the Lagrangian correspondence constructed between the relevant Hitchin moduli space and the Hilbert scheme of points on $T^*C$, and let the Dolbeault geomet…

Let $C$ be a compact connected Riemann surface, let $D$ be an effective divisor on $C$, and let $M_H$ and $M_{dR}$ denote the relevant Higgs and de Rham moduli spaces. Let…

Let $C$ be a compact connected Riemann surface of genus at least $2$, and let the Hitchin moduli spaces and their Lagrangian correspondences be as constructed in the paper. Let…

Let … be a $(-1)$-shifted exact oriented Lagrangian correspondence. Here $\phi_M$ and $\phi_N$ are the perverse sheaves of vanishing cycles associated with the oriented $(-1)$-shif…

The paper considers an immersed smooth quilt as depicted in Figure 22, with the corresponding quilted geometric data and boundary and seam conditions. Cazassus–Herald–Kirk–Kotelski…

Let $Y^+$ be a cylindrical-end $3$-manifold with boundary surfaces $\Sigma_-$ and $\Sigma_+$, let $\mathfrak{s}$ be a $\operatorname{Spin}^c$ structure, and let…

Let $Y$ be an elementary cobordism from $\Sigma_{g+1}$ to $\Sigma_g$ with a Morse function having one index-$1$ critical point, let $\mathfrak{s}_d$ be a degree-$d$…

Let $Y$ be a $3$-dimensional cobordism from $\Sigma_-$ to $\Sigma_+$, let $f$ be an adapted harmonic Morse function, let $Y^+$ be the associated manifold with cylindrical ends, and…

Functoriality conjecture. A Lagrangian correspondence in $X_0^- X_1$ provides a functor between the Fukaya categories of $X_0$ and $X_1$ which acts on objects by geometric composit…

Let $\mathrm{CompCorr}_{n}^{s,\operatorname{nd}}$ be the $(\infty,n)$-category of nondegenerate $s$-shifted compatible coisotropic correspondences, let…

Let $M_{0}$ be a Liouville manifold with Liouville sub-domain $U_{0} \subset M_{0}$, and let $M$ and $U$ denote their wrapped Fukaya categories, each consisting of a suitable count…

Let $(X_i,c9_i)$ be compact spin symplectic manifolds, and let $L_1$ and $L_{12}$ be spin immersed Lagrangian submanifolds of $(X_1,c9_1)$ and $(X_1X_2,-_1^*(c9_1)+_2^*(c9_2))$, re…

Joyce's conjecture. There should exist a morphism of constructible sheaves

Joyce's conjecture. Lagrangians in $X$ should define sections of the sheaf $\mathcal{P}_X$.

Figure-eight bubble conjecture. The moduli space of figure-eight bubbles should give a bounding cochain, in the sense of Fukaya–Oh–Ohta–Ono, for the filtered $A_{\infty}$ algebra a…

Let $M_0$, $M_1$, and $M_2$ be compact symplectic manifolds. Write $M_i^-$ for $M_i$ equipped with the opposite symplectic form, and let $\operatorname{Fuk}(M_1^-\times M_2)$,…

Let $M_0$, $M_1$, and $M_2$ be compact symplectic manifolds. Consider the Fukaya categories of cleanly immersed Lagrangians in $M_1^-\times M_2$, $M_0^-\times M_1$, and…