47 problems

A Calabi–Yau manifold is a Kähler manifold equipped with a nowhere-vanishing holomorphic volume form, and a Lagrangian submanifold is stable in an appropriate sense as specified by…

Let $K$ be a Lagrangian submanifold and let $\phi$ range over Hamiltonian isotopies, so that $\phi(K)$ ranges over the Hamiltonian isotopy class of $K$. The monotone Clifford torus…

Arnol'd's nearby Lagrangian conjecture. The Hamiltonian orbit of the zero section $L$ should be the set of all exact Lagrangians $L'$. This conjecture concerns the classification o…

Let an embedded annulus-type minimal Lagrangian surface with Legendrian capillary boundary be an embedded minimal Lagrangian surface in the unit ball $\mathbb B^4$ whose topology i…

Let $\omega_{\mathrm{FS}}$ denote the Fubini–Study form on $\mathbb{CP}^n$. For a closed Lagrangian torus $L\subset(X,\omega)$, call $L$ extremal when…

Let $0<a_1\le a_2\le\cdots\le a_n<\infty$, and let $E^{2n}(a_1,\dots,a_n)\subset\mathbb{C}^n$ be the ellipsoid defined by … For a closed Lagrangian torus $L\subset(X,\omega)$, set…

Centeredness conjecture. Every Hamiltonian stationary cyclic Lagrangian surface of type III is centered.

For compact Lagrangians $L,L'$ in a symplectic manifold $(M,\omega)$, let $B(L,L')$ be the supremum of the numbers $r a 0$ for which there exists a symplectic embedding … satisfyin…

Let $M$ be a Calabi–Yau manifold, and let a special Lagrangian cycle consist of a special Lagrangian submanifold equipped with a unitary flat connection. SYZ mirror conjecture. The…

Let $L$ be one of the special Lagrangian tori in ${\bf R}^4$ constructed by Y. Chekanov, with its Lagrangian projection to the relevant base. Chekanov's conjecture. The Lagrangian…

Sikorav's conjecture. In this standard basis, the class $[\iota^*\lambda]$ should be constrained to the positive orthant of $H^1(T^n;\mathbb{R})$. Chekanov proved that this conject…

Let $B^{2n}(1)$ be the symplectic unit ball, with $n\ge2$. Consider the capacity obtained by taking the supremum of $A_{\min}(L)$ over all closed Lagrangian submanifolds $L$ in the…

Let $0<a_1\le a_2\le\cdots\le a_n<\infty$, and let $E^{2n}(a_1,\dots,a_n)$ be the corresponding ellipsoid with standard symplectic form $\omega_{\mathrm{std}}$. A Lagrangian torus…

Let $(X,\omega,\Omega)$ be a compact Calabi–Yau manifold, and let $L_1,\dots,L_r$ be special Lagrangian submanifolds in $X$ with the same phase that intersect transversely. Let…

Let $K$ be an orientable polyhedral Lagrangian in $(\mathbb{R}^4,\omega)$, let $v$ be a valence-$4$ vertex, and let $\Gamma_{v,\delta}\subset (S^3,\xi_{\mathrm{std}})$ be the Legen…

Let $0<a_1\leq a_2\leq\dots\leq a_n<\infty$, and let $E^{2n}(a_1,a_2,\dots,a_n)$ be the corresponding ellipsoid with standard symplectic form. Ellipsoid extremal-torus conjecture.…

Let $X^{2n}_{\Omega}\subset \mathbb{C}^n$ be a convex toric domain, with standard symplectic form $\omega_{\mathrm{std}}$. Let $\operatorname{diagonal}(X^{2n}_{\Omega})$ denote the…

Let an extremal Lagrangian torus in $(\bar{B}^{2n}(1),\omega_{\mathrm{std}})$ mean a Lagrangian torus whose symplectic energy attains the relevant Lagrangian capacity. Cieliebak–Mo…

Pigati–Rivière regularity conjecture. Any Lagrangian stationary surface should be realised by a smooth branched immersion away from isolated Schoen–Wolfson conical singularities.

Let $X$ be a symplectic manifold, let $\bm L$ be a Lagrangian brane, and let $c_{\bm L}>0$ be as in the classification theorem. A sheaf quantization of $\bm L$ in $(X,u<c_{\bm L})$…

Let $X_{\Omega}$ be a convex or concave toric domain, let $c_L$ denote the Lagrangian capacity, and let $\delta_{\Omega}$ be the parameter associated with the domain in the source.…

Let $X$ be a Calabi-Yau manifold with Kähler metric. Let $L$ be an almost calibrated exact Lagrangian brane in $D^bFuk(X)$ with a Harder-Narasimhan decomposition … whose distinguis…

Let $\Bbbk$ be the coefficient field and consider the inclusion $\mathbb{RP}^2\subset\mathbb{CP}^2$. Real-projective-plane fullness conjecture. The inclusion is $SH$-full if…

In Tonkonog's setup, let $\mathcal{BS}\in SH^0(X;\Bbbk)$ be the class obtained by counting holomorphic caps, and write $\beta=\beta_1+\beta_2+\cdots$ for the filtration components…

Let $a\in QH^0(M;\Lambda)$ be the idempotent generating the ideal $QH^*(M;\Lambda)_{\mathrm{crit}}$, and let $\mathbb{L}$ be the skeleton. A subset is $a$-$SH$-full when its $a$-re…