27 problems

Let $\pi:W\to D$ be an exact symplectic Lefschetz fibration over the disk, and let $L_1,\dots,L_m$ be its vanishing cycles. Let the associated $A_\infty$-category be the category w…

Periodicity conjecture. The positive equivariant symplectic homology of $M$ is periodic.

Prequantization and minimal-orbit conjecture. Every such contact manifold is a prequantization of an orbifold, and the minimal number of simple closed Reeb orbits among its contact…

Finite-orbit prequantization conjecture. Any such contact form is necessarily non-degenerate and lacunary, meaning that the Conley–Zehnder indices of all closed orbits have the sam…

Let $W\subset\mathbb R^{2n}$ be a star-shaped domain and let $\operatorname{SH}^t(W;\mathbb F)$ denote its filtered symplectic homology with coefficients in a field $\mathbb F$. Th…

Local symplectic-homology equivariance conjecture. For every $k\in\mathbb{N}$, at least with rational coefficients, one has

Iterated Reeb SDM conjecture. The presence of an iterated Reeb SDM implies that the Reeb flow has infinitely many simple Reeb orbits.

Local homology concentration conjecture. The second condition in the theorem, namely the existence of an integer $d$ satisfying

Let $Q$ be a manifold and let $W$ be a Liouville domain. Write $\widehat{\mathbb{H}}_*^{S^1}\Lambda$ for Rabinowitz string homology, $S\mathbb{H}_*^{S^1}({\partial}W)$ for $S^1$-eq…

Let $Q$ be a manifold, let $W$ be a Liouville domain of dimension $2n$, and let $\widehat{\mathbb{H}}_*\Lambda$, $\overline{\mathbb{H}}_*\Lambda$, $S\mathbb{H}_*({\partial}W)$, and…

For $\Omega\in{\mathscr{S}}^2$, let $X_\Omega\subset{\mathbb C}^2$ be the associated star-shaped toric domain, and let $c_k(\Omega)$ be the chain-complex capacities defined from…

Let ${\mathscr{S}}^2$ denote the class of star-shaped regions used to define the filtered chain complex $C^\Omega_*$. For $\Omega\in{\mathscr{S}}^2$, set … For $0<a<b\leq\infty$, l…

Let $(M,omega)$ be a closed symplectic manifold, and let $K\subset M$ be compact. The subset $K$ is heavy if it satisfies the heavy-set condition defined using the relevant symplec…

Let $A$ be a convex body in $\mathbb R^{2m}$, let $c_{SH}$ be the symplectic homology capacity, and define … Write $B^{2m}(r)$ for the Euclidean ball of radius $r$. Weak Viterbo co…

Let $x=y^k$ be the $k$-th iteration of a simple Reeb orbit $y$, and let $f$ be the return map associated to $y$. The local contact homology of $x$ is denoted by $HC(x)$, and…

Let $(M_1,comega_1)$ be the compact part of a symplectic completion $M$, let $\Sigma$ be its contact-type boundary, and let … be the connecting homomorphism associated with the ide…

Let $W$ be a Liouville domain and let $V$ be obtained from $W$ by attaching a subcritical exact symplectic handle of index $k<n$ to $\partial W\times[0,1]$. Assume that the Conley–…

Bourgeois–Ekholm–Eliashberg cyclic surgery conjecture. Under this isomorphism, the surgery exact triangle for cyclic Legendrian contact homology is identified with the exact triang…

Let $W$ be a Liouville cobordism corresponding to attaching critical handles along a collection $\Lambda$ of Legendrian spheres, and let $L{\mathbb H}^{\text{Ho}}(\Lambda)_*$ and…

Bourgeois–Ekholm–Eliashberg surgery conjecture. These isomorphisms identify the tautological exact triangle involving $SH_*^{=0}$, $SH_*$, and $SH_*^{>0}$ for $(W,\partial^-W)$ wit…

The method under discussion constructs exotic spheres with the stated symplectic-homological properties in dimensions $11$ and $15$. Conjecture. This method can be applied in any d…

Compatibility conjecture. The two previous isomorphisms should be chosen so that the Gysin exact sequence and the Connes exact sequence are isomorphic. The claim proposes a compati…

Seidel's conjecture. The displayed isomorphism should hold. Seidel also implicitly conjectured the corresponding equivariant statement, identifying $SH_*^{S^1}(E)$ with…

Gysin–Connes compatibility conjecture. The two isomorphisms can be chosen so that the Gysin exact sequence for $S^1$-equivariant symplectic homology and the Connes exact sequence a…

Let a Lefschetz fibration have vanishing cycles defining a curved $A_\infty$-algebra $\mathcal{D}$. The wrapped Fukaya category is compared with the category of modules over…