27 problems

Joyce's conjecture. There exists a Bridgeland stability condition $\sigma=(Z,\mathcal{P})$ on $\operatorname{DFuk}(M)$ such that the central charge is induced by

Thomas's conjecture. There should be a stability condition for compact graded Lagrangians in a Calabi--Yau manifold that determines the existence and uniqueness of a special Lagran…

Let a complex structure determine a Fukaya category, and let special Lagrangians be the relevant geometric objects in that category. Stability conjecture. Complex structures are st…

Let $L$ be a stable Lagrangian for which the phase-closeness condition fails, so that mean curvature flow may become singular in finite time and split $L$ into pieces. Smoothness c…

Let $\check{\mathcal{X}}$ be the mirror geometry, let $L$ be a Lagrangian in $\check{\mathcal{X}}$, and let $\mathcal{E}$ be a holomorphic vector bundle on $\mathcal{X}$. A Bridgel…

Joyce's conjecture. The flow exists for all times and converges to a sum of special Lagrangian integral currents, which is isomorphic to the Harder–Narasimhan decomposition of…

Joyce's decomposition conjecture. Any Lagrangian with unobstructed Floer homology can be decomposed in a suitable sense into special Lagrangians, using the Lagrangian mean curvatur…

Folklore product conjecture. The form $\Omega_{\mathbf{a}^{(i)}}$ induces a stability condition on the Fukaya–Seidel-type category mirror to $\operatorname{D^b}(X_i)$, and stable o…

In a $G_2$ manifold equipped with a Kovalev–Lefschetz fibration, gradient cycles are cycles in the base playing the role of tropical curves for the enumeration of associative subma…

A zero-Maslov Lagrangian is a smooth Lagrangian submanifold whose mean-curvature one-form has trivial cohomology class. Consider zero-Maslov Lagrangian mean curvature flow in a Cal…

A Calabi–Yau manifold is considered in the large complex structure limit, and its special Lagrangian torii have a moduli space equipped with a Hessian metric and an associated affi…

Let $X$ be a Calabi–Yau $3$-fold with a Lefschetz K3-fibration $\pi:X\rightarrow Y$, and let a gradient cycle be a graph in $Y$ whose edges are decorated by $2$-cycles in the K3 fi…

Let $X$ be a Calabi–Yau $3$-fold with a K3-fibration $\pi:X\rightarrow Y$, let $\omega_X$ and $\omega_Y$ be Kähler forms on $X$ and $Y$, respectively, and let…

Let $X_t$ be a $1$-parameter family of compact Calabi–Yau manifolds with metrics $g_t$ admitting special Lagrangian fibration structures and with bounded diameters, and let $C_t$ b…

Let $X$ be a Calabi-Yau manifold, and let $L$ and $L'$ be special Lagrangian integral currents with the same phase angle $\hat{\theta}$. Equip them with suitable unobstructed brane…

Let $\mathcal{L}$ be the admissible class of Lagrangian currents and let $\mathcal{S}$ be the Solomon functional. Let $\hat{\theta}$ be the phase determined by the relevant Fukaya-…

Let $\mathcal{L}$ be the class of admissible Lagrangian objects and let $\mathcal{S}$ be the Solomon functional. A minimizer is an element $L\in\mathcal{L}$ attaining the infimum o…

Let $\mathcal{L}$ be the class of quantitatively almost calibrated exact unobstructed Lagrangian objects under consideration, and let $L_0\in\mathcal{L}$ be exact, quantitatively a…

Let $(X,\omega,\Omega)$ be an exact Calabi-Yau manifold, and let $L_0$ be a nontrivial unobstructed exact Lagrangian brane with quantitatively almost calibrated phase function. Let…

Let $X$ be a compact almost Calabi-Yau manifold. Let $L_0$ be a reference Lagrangian, and consider graded immersed unobstructed Lagrangians in the same $D^bFuk(X)$ class as $L_0$.…

Let $X$ be a Calabi-Yau manifold, let $D^bFuk(X)$ denote its derived Fukaya category, and let Joyce's conjectural Bridgeland stability be a stability condition on this category. An…

Let $X$ be a Calabi–Yau manifold, let $L\rightarrow X$ be a holomorphic line bundle, and let $L\hookrightarrow X$ be a Lagrangian submanifold. The objects $L$ are viewed respective…

Let $L$ be a Lagrangian in a Calabi--Yau manifold. Thomas--Yau conjecture. A suitable notion of stability, called flow stability, should imply long-time existence and convergence o…

A Fukaya category is associated with a symplectic manifold, and a stability condition is a structure whose stable objects and Harder–Narasimhan filtrations organize its objects. A…

Let $L$ be a Lagrangian submanifold of a Calabi–Yau manifold. Say that $L$ is stable when it is a stable object in the derived Fukaya category with respect to an appropriate Bridge…