53 problems

Let $N$ be a closed manifold, let $g$ be a Riemannian metric on $N$, and let $D_1T^*N=\{(q,p)\mid |p|_g\leq 1\}$. Let $\mathfrak L(T^*N)$ denote the relevant class of closed exact…

Let $X$ be a symplectic cluster $4$-manifold equipped with a cluster presentation $\{\pi,L_1,\dots,L_n\}$. A fiber means a fiber of $\pi$ over a point of the base. Non-displaceabil…

Let $s,s_i\in(1/2,1)$, let $k_i$ be positive even integers for $i=1,\dots,l$, and assume $n=\sum_i k_i$. Consider the Lagrangian tori $\Theta^n_s$ and…

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…

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

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

Let $L\subset M$ be a Maslov $0$ Lagrangian torus, and let $C^*(L,M)$ be its Floer complex with differential satisfying … Here $\tau(L)$ denotes the relevant boundary-depth invaria…

Giroux–Pardon conjecture. There is a Weinstein Lefschetz fibration

Let $B^{2n}(a)$ be the symplectic ball of capacity $a$, let $L$ be a Lagrangian submanifold in $B^{2n}(a)$, and let $J$ be an admissible almost-complex structure. Holomorphic-disk…

Hodge lift conjecture. For any complex Lagrangian $L$, the object

Microsheaf wrapping conjecture. There exists an inductive system $\{\mathcal E_i\}$ in $\mu\operatorname{sh}_{\mathbb C\text{-}c}(X)$ such that

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 $L$ be a displaceable Lagrangian in a symplectic manifold $M$, and let $d_{C^0}$ be the $C^0$ distance on Hamiltonian diffeomorphisms. Local Hamiltonian nondisplacement conject…

Let $M$ be a compact or Liouville symplectic manifold, and let $L\subseteq M$ be a compact, relatively exact Lagrangian. Let $\tilde\phi:L\to L$ be a diffeomorphism extending to a…

Let $N$ be a manifold, let $T^*N$ carry the cotangent-bundle structure, and let … DrT^N={(q,p)in T^N:|p|leq r}. … is bounded on the space of exact Lagrangians contained in…

Let $X$ be a closed Kähler manifold, and let $\Sigma\subset X$ be a complex hypersurface such that $X\setminus\Sigma$ is a Weinstein manifold admitting a cyclic dilation; then…

Let $V$ be the Liouville sector and let $\mathcal L_\gamma$ and $\mathbb{T}_\gamma$ be the Lagrangian cylinder and the torus obtained from it by wrapping and resolving the resultin…

Let $Q_{d53f\mathbb{R}P^2}$ denote the matrix factorization corresponding, under homological mirror symmetry, to the monotone Lagrangian $(\mathbb{R}P^2,b=0)$, and let … Here…

Superabundance and non-wide conjecture. The tropical curve $V$ is superabundant if and only if its Lagrangian lift $L_V$ is not wide.

Fiberwise isotopy conjecture. The subspace of $H^1(L_V,\mathbb R)$ arising from flux classes of fiberwise Lagrangian isotopies is identified with…

Unobstructedness and B-realizability conjecture. The geometric Lagrangian lift $L_V$ is unobstructed if and only if $V$ is $B$-realizable.

Let $N$ be a closed Riemannian manifold and let $DT^*N=\{(q,p)\in T^*N\mid |p|_g\leq 1\}$. For an element $L\in\widehat{\mathfrak L}(T^*N)$, its $\gamma$-support is denoted by…

Let $N$ be a closed Riemannian manifold, and let … An exact Lagrangian $L$ is geometrically bounded when it is contained in $DT^*N$. Geometrically bounded implies spectrally bounde…

Finite monodromy classification conjecture. Either $\mathcal{H}_L$ is isomorphic to a subgroup of

Let $L$ be the total space of the fibration defined by the theorem preceding the conjecture, with $k=1$, $p_1=4$, $p_2=11$, and $n=20$. Thus $L$ is the specific monotone Lagrangian…