107 problems

Let $N$ be a closed manifold, let $T^*N$ be its cotangent bundle, and let $\mathcal{O}_N$ denote the zero section. Write $\mathscr{L}(T^*N)$ for the space of closed exact Lagrangia…

A Lagrangian torus is a Lagrangian submanifold $L\subset\mathbb{C}^n$ diffeomorphic to a torus. A disk with boundary on $L$ has Maslov index given by its Maslov class evaluated on…

Floer-theoretic identification conjecture. The Floer-theoretic coset is the same as the nearby Lagrangian torsion:

Let $(X,\omega)$ be a symplectic manifold, let $\psi\in\operatorname{Ham}(X,\omega)$, and let $L\subset(X,\omega)$ be a closed Lagrangian submanifold. Lagrangian Poincaré recurrenc…

Let $(W,d\theta)$ be a Weinstein manifold and let $L\subset W$ be an exact closed Lagrangian. A regular Lagrangian is an exact closed Lagrangian that is a subset of the skeleton fo…

Let $M$ be a closed connected manifold, and let $\mathcal{L}(M)$ be the space of closed connected exact Lagrangian submanifolds of $T^*M$. Strong nearby Lagrangian conjecture. The…

Biran–Cornea's conjecture. Every monotone Lagrangian $L\subset\mathbb{C}P^n$ is either wide or narrow. This predicts a dichotomy for the quantum cohomology of monotone Lagrangians;…

Let $X$ be the symplectic four-manifold under consideration, let $\mathcal P:{\mathcal L}(X)\to {\rm Sh}_{H}^+(X)$ be the projection assigning to a Lagrangian torus its area class,…

Let $P_1$ and $P_2$ be Delzant Fano polytopes in $\mathbb{R}^k$ with $n+k$ facets, and let $L_1$ and $L_2$ be the corresponding embedded monotone Lagrangian submanifolds of…

Let $L\subset(\mathbb{C}P^2,\omega_{\operatorname{FS}})$ be an embedded monotone Lagrangian two-torus, and let $\Lambda$ be the Legendrian lift of its canonical threefold Bohr–Somm…

Let $L$ and $L'$ be closed exact Lagrangian submanifolds of a convex exact symplectic manifold $X$. Let $HF^*(L,L')$ denote their Floer cohomology, and let their Whitehead torsion…

Let $X$ be an irreducible holomorphic symplectic manifold of $K3^{[n]}$-type, meaning that it is deformation equivalent to a Hilbert scheme of $n$ points on a $K3$ surface. Let…

Let $M \to B$ and $\check{M} \to B$ be dual torus fibrations arising in the SYZ picture, and let a Lagrangian section of $M \to B$ be a Lagrangian submanifold mapping isomorphicall…

Let $L$ be a compact Lagrangian submanifold embedded in $cC^n$, and let its Maslov class be denoted by $\mu_L$. Maslov Class Conjecture. The Maslov class is nonzero: … This is pres…

Let $L$ be a monotone Lagrangian submanifold with minimal Maslov number $\Sigma(L)\geq 3$, let $e_H(L)$ denote Hofer's symplectic energy, let $k(L,\phi)$ be the invariant associate…

Let $(M,omega)$ be a closed symplectic manifold and let $L,L'$ be closed Lagrangian submanifolds. Writing the left-hand inequality in Corollary i. as the inequality involving the q…

Let $M$ be a smooth manifold and let $T^*M$ denote its cotangent bundle with the canonical symplectic structure. A Lagrangian submanifold $L\subset T^*M$ is exact if the canonical…

Oh's conjecture. The fixed point set of $\tau$ is globally volume minimizing under Hamiltonian deformations.

Fomenko's conjecture. All Maslov–Arnold characteristic classes of minimal Lagrangian surfaces are equal to zero.

Let $(M,\omega)$ be a compact symplectic manifold and let $L$ be an oriented Lagrangian submanifold of half the dimension of $M$, so that $\omega|_L=0$. For a generic almost-comple…

Let $\Lambda^0=S^1\times\mathbb{R}P^n$ be the constant loop of the Lagrangian submanifold $\mathbb{R}P^n$. The invariant $\varepsilon(\Lambda^0)$ is defined for this loop. Constant…

Lagrangian–sheaf conjecture. Fundamentally, mirror symmetry is a relation between the Lagrangian submanifolds of a threefold and the semistable coherent sheaves on its mirror.

Let $L$ be a monotone Lagrangian submanifold with $\Sigma (L) \geq 3$, let $e_H(L)$ denote its Hofer symplectic energy, and let $\sigma (L)$ be the minimal symplectic action on…

Generation conjecture. The specified collection of compact Lagrangians generates

Nullhomotopy-dependence conjecture. The classes $\tau_{2k}(L,\gamma_t^i)$, for $i=1,2$, differ by $p_k(E)$.