552 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 $X$ be a closed $2n$-manifold and let $\omega$ be an integral symplectic form on $X$, meaning that its cohomology class lies in the image of…

Let $Z$ be the open Batyrev–Borisov complete intersection with skeleton … where $S=\partial\widetilde{C}_{0,\beta}\subset M_\R$ is the codimension-$r$ sphere introduced above, and…

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…

Smooth-to-holonomic equivalence conjecture. The inclusion

Sheaf quantization and Fukaya category conjectures. For each Lagrangian brane $\bm L$ in $X$, there is an almost quasi-isomorphism

Ambient-monodromy realization conjecture. An isomorphism $\Phi\in\operatorname{Aut}H_2(X,T(x))$ can be realized as the ambient monodromy of a Hamiltonian diffeomorphism mapping…

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 $V$ be a linear representation of a complex reductive algebraic group $G$ and let … be the associated moment map. Set $V:=\mu^{-1}(0)$. For a generic…

Viterbo's conjecture.

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…

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 $n\in\mathbb{Z}_{\geq 2}$ and define … Let $c_{\beta_n}$ denote the capacity function associated with the four-dimensional polydisk, and let $\operatorname{accum}(\beta_n)$ den…

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 $D_n$ be the circular spherical divisor with $n$ components discussed above. A rational embedding means an embedding of $D_n$ into a rational surface. Rational embeddability co…

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 $X$ be the bordered holomorphic curve and let $\tilde M(f)$ be the compactified moduli space associated with a fixed homotopy class $[f]$. A simple path in $X$ has no self-inte…

Let $M$ be a manifold with symplectic form $\omega$ and compatible almost complex structure $J$, let $L_1(j),\dots,L_{p_j}(j)$ be transversely meeting Lagrangian submanifolds, and…

Let $J_1$ and $J_2$ be choices of almost complex structure, and let ${\Bbb R}_1$ and ${\Bbb R}_2$ be the corresponding elements of the completed exterior-algebra construction. Let…

Let $R^k(\alpha)=\sum_{g=0}^{\infty}R^k_g\alpha^g$ be the formal power series whose coefficients are defined by counts of holomorphic maps from a genus-$g$ curve with $k$ boundary…

Let two spheres in $\mathbb{R}^3$ be equipped with oriented normal bundles and intersect along a circle in the space $\tilde Y$, with the canonical orientation induced on the inter…

Let $\mu:M\to\mathfrak{g}$ be proper, let $0$ be a regular value of $\mu$, assume that $\mu^{-1}(0)$ is nonempty, and suppose that $\mathrm{G}$ acts freely on $\mu^{-1}(0)$. For…

Symplecticity conjecture. For sufficiently large $k$, the transverse hyperplane sections of $F_k$ are symplectic submanifolds of $X$.

A Moishezon 4-manifold is the smooth 4-manifold constructed from a semi-algebraic cuspidal curve in $\mathbb P^2$ and an epimorphism to a symmetric group, with a smooth map to…