1,184 problems

Let $N$ be a closed manifold, let $T^*N$ be its cotangent bundle, and let $\mathcal{O}_N$ denote the zero section. Write $\mathscr{L}(T^*N)$ for the space of closed exact Lagrangia…

Atiyah–Floer conjecture. Under appropriate hypotheses, there should be an isomorphism

Brylinski's conjecture. Every $k$-de Rham cohomology class contains a symplectic harmonic representative.

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…

Higgs branch conjecture. The Higgs branch, viewed as a holomorphic symplectic variety, is canonically isomorphic to the associated variety of the corresponding VOA:

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 $A_n(\mathbb{C})$ be the $n$-th Weyl algebra, and let $\mathbb{C}^{2n}$ carry its standard Poisson structure. A polynomial symplectomorphism of $\mathbb{C}^{2n}$ is a polynomia…

Let $(X,\lambda)$ be a Weinstein Liouville domain with skeleton $\skel(X,\lambda)$. Denote by $\mW(X,\lambda)$ its wrapped Fukaya category. Kontsevich's cosheaf description conject…

Let $M$ be a compact pre-quantizable symplectic manifold admitting a Hamiltonian action of a compact connected Lie group $G$. Guillemin–Sternberg's conjecture. Quantization commute…

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…

Let $(M,\omega)$ be a compact symplectic manifold, and let a Hamiltonian map on $M$ have more fixed points than are necessarily required by the V. Arnold conjecture. Hofer-Zehnder…

A Lagrangian torus is a Lagrangian submanifold $L\subset\mathbb{C}^n$ diffeomorphic to a torus. A disk with boundary on $L$ has Maslov index given by its Maslov class evaluated on…

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

Teleman's conjecture. A Hamiltonian $G$-action on $X$ should be mirror to a holomorphic fibration

Arnol'd's conjecture. Every Hamiltonian diffeomorphism $\varphi$ possesses at least as many fixed points as a smooth function $f:M\to\mathbb{R}$ possesses critical points.

Let $(M,\omega)$ be a positive monotone symplectic manifold of dimension six admitting a Hamiltonian circle action. A Fano manifold is a compact complex manifold whose anticanonica…

Let $(M,g)$) be a closed Riemannian manifold. Define … and let $0_M$ denote the image of the zero section of $T^*M$. For a compactly supported Hamiltonian diffeomorphism…

Rabinowitz's minimal periodic solution conjecture. Under these hypotheses, for every prescribed positive period, the system possesses a non-constant solution having that period as…

Let $(X,\omega)$ be a symplectic manifold equipped with a Lagrangian torus fibration admitting a Lagrangian section. Kontsevich–Soibelman associate to this data a rigid analytic sp…

Let $c_k^{GH}$ denote the $k$-th Gutt–Hutchings capacity and let $c_k^{EH}$ denote the $k$-th Ekeland–Hofer capacity. Gutt–Hutchings conjecture. For every $k=1,2,3,\ldots$, the two…

Landsman quantization reconstruction conjecture. The classical limit functor is almost inverse to the quantization functor defined by Landsman, in the sense that the classical limi…

Symplectic invariance conjecture. The free energies $\mathcal{F}^g$ are invariant under any symplectic transformation of $(x,y)$.

Let $d$ be an integer, and let $C$ be an algebraic curve of degree $d$ in $\mathbb{C}P^2$. Its genus is $\frac{(d-1)(d-2)}{2}$. Thom's conjecture. The genus of $C$ is minimal among…

Flux Conjecture. The subgroup $\operatorname{Ham}(M,\omega)$ is $C^1$ close in $\operatorname{Symp}(M,\omega)_0$; equivalently, it is $C^1$-closed there.

Stipsicz's conjecture. Every fiber-sum indecomposable Lefschetz fibration admits an exceptional section. Equivalently, every Lefschetz fibration should be a fiber-sum of blown-up p…