18 problems

Arnold–Givental conjecture. The number of intersection points of $\varphi(L)$ and $L$ is at least the total dimension of the cohomology of $L$.

Let $(M,\omega)$ be a closed symplectic manifold, let $L$ be a closed Lagrangian submanifold of $M$, and let $\varphi\in \operatorname{Ham}(M,\omega)$ be a Hamiltonian diffeomorphi…

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

Point-plus-connected clean-intersection conjecture. The only such example is $C=\mathbb{RP}^{n-1}$, at least up to homotopy-equivalence, and one always has $\mu(p)\in\\{0,n\\}$.

Clean-intersection homotopy conjecture. The manifold $C$ must be at least homotopy-equivalent to $\mathbb{RP}^{n-1}$.

Arnol'd Conjecture. Under these assumptions, the lower bound

Let $C\subset \mathbb{R}^n$ be a convex body with smooth boundary. For $q\in C$, define $L_q=\operatorname{graph}({\rm d}\langle q,-\rangle)\subset T^*S^{n-1}$, and let…

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…

The cuplength Lagrangian Arnold conjecture. Without assuming that $L$ and $\varphi(L)$ intersect transversely, the number of intersection points is at least $cl(L)$. The source say…

Let $(M,\omega)$ be a connected tame symplectic manifold, let $L$ be a compact Lagrangian submanifold, let $\varphi=\varphi_H^1$ be a Hamiltonian diffeomorphism, and let…

Let $(M,c?\omega)$ be a connected tame symplectic manifold, let $L$ be a compact Lagrangian submanifold, and let $\varphi=\varphi_H^1$ be a Hamiltonian diffeomorphism. The strong L…

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

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 $(M,\omega)$ be a compact symplectic manifold and let $L\subset M$ be an exact Lagrangian submanifold, meaning that the restriction of a primitive of $\omega$ to $L$ is exact.…

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…