560 problems

Let $S(2n,\mathbb{R})$ denote the space of real symmetric $2n\times 2n$ matrices, let $J$ be the standard symplectic matrix, and let $\rho$ be the symplectic-group action on…

Let $p$ and $q$ be monic polynomials of degree $2$, let $V$ be a vector space, and let $(b,u)$ be a symplectic pair on $V$. A symplectic $(p,q)$-difference is a symplectic pair sat…

Let $(W^4,\vlambda,\vphi)$ be a $4$-dimensional Weinstein domain, and let $L\subset W$ be a properly embedded Lagrangian disk with Legendrian boundary…

Determine whether celestial-mechanics systems possess universal Poincaré maps: namely, whether, for every symplectic embedding $f$ of the disk into $\mathbb{R}^{2}$ and every…

For every integer $n\geq 2$, let $F\colon B^n\to B^{2n}\subset\mathbb{C}^n$ be a smoothly immersed Lagrangian $n$-ball whose Maslov form is conformal and whose boundary is Legendri…

For every dg algebra $A$ over a field $k$ such that $\dim_k H^*(A)<\infty$, the regular module $A$ cogenerates the unbounded derived category $D(A)$ of dg $A$-modules; equivalently…

In the appropriate Fukaya-category setting for a compact Calabi–Yau manifold $X$, if a Lagrangian submanifold $L\subset X$ defines an object $E_L$ that is stable with respect to th…

Let $\Sigma_g$ be a closed oriented surface of genus $g\ge 2$, let $\Lambda$ be the complex Novikov field, and let $\mathscr F_g$ be the split-closed, strictly unobstructed, two-pe…

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with real-analytic boundary containing the segment joining the two centres, and let $B_h$ denote the billiard map on the fixed-e…

For every admissible lattice point $(\chi,c)$, does there exist a closed, simply connected, spin symplectic $4$-manifold $X$ such that $(\chi_h(X),c_1^2(X))=(\chi,c)$?

Let $N$ and $L$ be closed, simply connected manifolds of the same dimension. The cotangent bundle $T^*N$ carries its natural symplectic structure, and consider the space of Lagrang…

Let $M=H_g^1\cup_{\Sigma_g}H_g^2$ be a Heegaard decomposition of a homology $3$-sphere, and let $I(M)$ denote its instanton Floer homology. For the trivial $SU(2)$ bundles on the h…

Polterovich's conjecture. There exists a constant $C_M$, depending only on the symplectic manifold $(M,\omega)$, such that for every finite open cover $\mathcal{U}=\{U_i\}$ of $M$,

Let $X$ be a smooth Fano variety of complex dimension $r$, let $g$ be a genus, and let $A$ be a curve class. The fixed-domain Gromov–Witten invariants are the virtual counts associ…

Let $L\subset S^3$ be a link. The singly graded symplectic Khovanov cohomology is denoted by $Kh^k_{symp}(L)$, and the bigraded combinatorial Khovanov homology by $Kh^{i,j}(L)$. Se…

Viterbo's conjecture.

Arnold's conjecture for non-degenerate Hamiltonians.

Tian–Jun Li's symplectic cone conjecture.

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…

Arnold's conjecture. The number of fixed points of $f$ satisfies

Let $(M,\omega)$ be a compact symplectic manifold of real dimension $4$, equipped with an almost complex structure $J$ tamed by $\omega$. Let $\tilde{\omega}$ be another symplectic…

Let $X$ be a Calabi–Yau manifold. A Lagrangian fibration is a map $\pi:X\to B$ onto a topological manifold $B$ whose fibers are graded with respect to a holomorphic volume form…

Let $X=\mathbb{C}P^2\# n\overline{\mathbb{C}P^2}$ carry a symplectic form $\omega$, and let $\Gamma(X,\omega)$ be its Lagrangian root system. Write $PR[\Gamma(X,\omega)]$ for the p…

Streets–Tian conjecture. Any closed Hermitian-symplectic complex manifold $(X,J)$ admits a Kähler metric.

Hochschild cohomology quotient conjecture. There exists a $\mathbb{C}[s]$-module structure on $HH^*(\mathcal{W}(X))$ such that $HH^*(\mathcal{W}(\underline{X}))$ is isomorphic to t…