9 problems

Crofton-type conjecture. There exist a constant $C>0$ and a smooth function $\rho:M\to\mathbb{R}_{>0}$, depending only on $\mathcal{D}$ and $L$ and satisfying

Degenerate Arnol'd conjecture. The intersection satisfies

Let $L$ be a closed smooth manifold and let $T^*L$ denote its cotangent bundle. For every compactly supported Hamiltonian diffeomorphism $\phi$ of $T^*L$, consider the intersection…

Let $(X,\omega)$ be a symplectic manifold, and let $L$ and $L'$ be Lagrangian submanifolds of $X$. Assume that $X$ is either closed or a Liouville manifold, and that $L$ is connect…

Let $L$ be a compact nondisplaceable monotone Lagrangian submanifold of a tame symplectic manifold $(M,\omega)$, let $\varphi\in\mathcal{H}am_c(M,\omega)$, and let $\mathbb{F}$ be…

Weak Arnold–Givental conjecture. The Lagrangian submanifold $L$ is Hamiltonianly nondisplaceable.

Arnold–Givental conjecture. Assume that $(M,\omega,\rho)$ is a closed real symplectic manifold and $\phi_H$ is a Hamiltonian diffeomorphism such that $L$ and $\phi_H(L)$ intersect…

Let $L_1$ and $L_2$ be exact Lagrangians with Floer homology $HF(L_1,L_2)$ and Floer complex $CF(L_1,L_2)$, whose Whitehead torsion is denoted by $\tau(CF(L_1,L_2))$. Fukaya's conj…

Let $X$ be a smooth algebraic symplectic variety, and let $Y,Z\subset X$ be smooth Lagrangian submanifolds. Write $H\!H^{\bullet}(\mathscr{C})$ for the Hochschild cohomology of a t…