18 problems

Let $W$ be a $C^1$ Lagrangian torus invariant under the geodesic flow of a Riemannian metric on $\mathbb{T}^n$, containing $n$ orbits whose canonical projections have well-defined,…

Let $H:T^*\mathbb{T}^n\to\mathbb{R}$ be a Tonelli Hamiltonian, and consider Lagrangian invariant tori at supercritical energy levels. A Hedlund Lagrangian torus is one for which th…

Uniqueness conjecture for toroidal generators. Any two toroidal generators of a toroidal subcategory are quasi-isomorphic, up to a shift.

For each integer $m\geq 2$, let $(Y_m,\omega_m)$ be the toric symplectic manifold obtained from $\mathbb{CP}^{2}$ with lines of area $m+2$ by symplectically blowing up along the st…

Let $L_{1,1}$ denote the Clifford torus in $(\mathbb{CP}^{2},\omega_{\mathrm{FS}})$, and let an integral packing mean a packing by disjoint monotone Lagrangian tori whose associate…

Let $d4c$ be a Plücker sequence of type $(k,n)$, and let $L_\mathfrak{s}\subset\operatorname{Gr}(k,n)$ be its associated Plücker torus. The Laurent polynomial $W_\mathfrak{s}$ is d…

For each Markov triple $(a,b,c)\in\mathfrak M$, let $T_{a,b,c}\subset\mathbb{C}P^2$ be Vianna's monotone Lagrangian torus, and let ${\frak w}_{\text{DW}}(T_{a,b,c};\mathbb{C}P^2)$…

Let $T_{a,b,c}\subset\mathbb{C}P^2$ be Vianna's torus associated with a Markov triple $(a,b,c)$, let $\mathbb{R}P^2\subset\mathbb{C}P^2$ be the real projective plane, and let…

Let $T_{a,b,c}\subset \mathbb{C}P^2$ be Vianna's monotone Lagrangian torus associated with a Markov triple $(a,b,c)\in\mathfrak M$. A capacity conjecture. There is no monotone symp…

Folklore conjecture. Every non-monotone Lagrangian torus in $\mathbb{R}^4$ is Hamiltonian isotopic to a product torus.

Almost-toric-fibre isotopy conjecture. Any Lagrangian torus in $\mathbb{C}P^2$ is isotopic to some $(a,b,c)$-ATF fibre.

Torre's conjecture. Any monotone Lagrangian torus in $\mathbb{C}P^2$ is of the form $T(a^2,b^2,c^2)$, up to symplectomorphism.

Let $\mathbb{C}P^n$ carry the Fubini–Study symplectic form $\omega_{\operatorname{FS}}$. A Lagrangian torus $L\subset \mathbb{C}P^n$ is extremal when it realizes the relevant Lagra…

Let $D^{2n}$ be the $2n$-dimensional unit disc with the standard symplectic form $omega0$, and let $S^{2n-1}=partial D^{2n}$. A Lagrangian torus $Lsubset D^{2n}$ is extremal w…

Let $T^c_{r,0}$ be a Lagrangian torus in $\mathbb{CP}^1\times\mathbb{CP}^1$ of type $T(1,2,9)$, and let a family of Maslov-index-$2$ holomorphic discs mean a family of such discs w…

Let $S_b$ be a regular fiber of a real polarization of $\mathbb{C}\mathbb{P}^2$, and let $p_{d_1,d_2}$ be the lifted period map. The diagonal associated with $(d_1,d_2)$ is the loc…

Let $S_b=\pi^{-1}(b)$ be a regular Lagrangian fiber of the real polarization of $\mathbb{C}\mathbb{P}^2$, and let $p_{d_1,d_2}=(p^1_{d_1,d_2},p^2_{d_1,d_2})$ be its lifted period m…

Let $S_b=\pi^{-1}(b)$ be a regular Lagrangian fiber over $b\in B\setminus\partial B$, and let $p_{d_1,d_2}=(p^1_{d_1,d_2},p^2_{d_1,d_2})$ be the lifted period map associated with g…