12 problems

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…

Generic singularity conjecture. In two dimensions, neck pinches and teardrops are the only singularities that appear generically.

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…

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…

Thomas–Yau conjecture. If the flow has a finite-time singularity, then $L$ can be decomposed into a graded connected sum $M_1\# M_2$ satisfying

Consider an equivariant Lagrangian mean curvature flow in the setting where a finite-time singularity is studied by type I rescaling. A type I blow-up is the limiting object obtain…

Lagrangian mean curvature flow can exhibit finite-time singularities followed by surgery that changes the Hamiltonian-isotopy type of a Lagrangian torus, such as the observed trans…

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…

Consider a type II singularity of a Lagrangian mean curvature flow, and let its type II smooth blow-up be an eternal exact, zero-Maslov solution. Steady-soliton conjecture. Type II…

Let a Lagrangian mean curvature flow start from embedded initial data, and suppose that it becomes immersed before its singular time. Embeddedness-loss conjecture. The flow must ha…

Let $T_1,T_2,\ldots$ be the candidate singular times associated with a Lagrangian mean curvature flow as in Theorem C. First-candidate conjecture. A type I singularity can occur on…

Let $M$ be a Calabi–Yau manifold and let $\Sigma$ be a compact embedded Lagrangian submanifold with zero Maslov class. Simplified Thomas–Yau conjecture. The mean curvature flow of…