85 problems

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…

Arnold–Givental conjecture. The number of intersection points of $\varphi(L)$ and $L$ is at least the total dimension of the cohomology of $L$.

Floer-theoretic identification conjecture. The Floer-theoretic coset is the same as the nearby Lagrangian torsion:

Let $f:\mathbb{C}^n\to\mathbb{C}$ be an algebraic map with an isolated singularity at the origin. Let $\mathcal{X}_m$ be the arc space of $f^{-1}(0)$, namely the space of maps from…

Let $D$ be a Liouville domain whose boundary $partial D$ has no contractible Reeb orbits. The intrinsic symplectic cochains $SC^*(D)$ and the $S^1$-equivariant symplectic cochains…

Let $L$ and $L'$ be closed exact Lagrangian submanifolds of a convex exact symplectic manifold $X$. Let $HF^*(L,L')$ denote their Floer cohomology, and let their Whitehead torsion…

There are two approaches to the study of $3$-dimensional gauge theory: the Seiberg–Witten–Floer approach, which counts solutions to the Seiberg–Witten equation, and the Heegaard–Fl…

Let $W$ be the fixed-point open subset associated with the involution on $\mathcal{Y}_{m,\tau}$, and let ${{\mathbb{T}}_{\hat{{\bf \alpha}}}}$ and…

Let $\mathcal{G}$ be the set of generators of the Seidel–Smith cochain complex, with gradings $J$ and $P$, where $J$ is the Jones grading and $P$ is the projective grading. Grading…

Continuity conjecture. In general, $A(c6;1)$ is continuous in the $C^c3$-topology, perhaps even in the $C^0$-topology, and therefore extends to arbitrary smooth Hamiltonian diffeom…

Let $Z$ be the space in the paper, let $\mathfrak{C}_*$ be the filtered chain complex constructed there, and let $C_*(Z)$ denote the cubical singular chain complex of $Z$, equipped…

Let $\mathfrak{C}_*$ and ${\mathscr C}_*$ be the filtered chain complexes constructed in the paper, and let the map $\XiBar$ be the map referred to in the source. Filtered homotopy…

Let $\{(f_t,g_t)\mid t\in[0,1]\}$ be a family of Morse data, and let $\Phi'$ be the bifurcation chain map and $\Phi$ the continuation map defined by counting flow lines of…

Let $c,c'$ be matching paths, let $\Lambda_c$ and $\Lambda_{c'}$ be the associated eventually Lagrangian submanifolds, and let $\mathcal A,\mathcal A'$ be directed $A_\infty$-categ…

Cohen's conjecture. The quantum field theory of Chas–Sullivan on $M$ is isomorphic to the Floer quantum field theory of $T^*M$.

Let $L$ be a monotone Lagrangian submanifold with $\Sigma (L) \geq 3$, let $e_H(L)$ denote its Hofer symplectic energy, and let $\sigma (L)$ be the minimal symplectic action on…

Let $S^n$ be the round $n$-sphere, let $\tau$ be its antipodal map, and define the twisted loop space … Let $\phi^d_\infty$ denote the wrapped Dehn twist on $T^*S^n$ obtained by co…

Nullhomotopy-dependence conjecture. The classes $\tau_{2k}(L,\gamma_t^i)$, for $i=1,2$, differ by $p_k(E)$.

Let $M$ be a del Pezzo surface of degree $1$, $5$, or $7$, and let $D$ be a generic smooth representative of $M$'s anticanonical divisor. A stable $\mathbb{R}$-polarization of $X$…

Let $\mathcal{S}$ be an abstract wrapped Floer setup, let $\mathcal{F}^{\text{pre}}_{\mathcal{S}}$ be its pre-wrapped Fukaya $A_\infty$-category, let $\widetilde{C}$ be the set of…

Let $(M,\omega)$ be a compact symplectic manifold, and let $\gamma$ be the Floer-theoretic pseudometric on the universal cover of $\mathrm{Ham}(M)$. Assume that $\omega$ vanishes o…

Maurer–Cartan conjecture for the family Floer curvature. For a suitable model of Floer theory, $\mu^0$ satisfies

Let $L\subset M$ be a Maslov $0$ Lagrangian torus, and let $C^*(L,M)$ be its Floer complex with differential satisfying … Here $\tau(L)$ denotes the relevant boundary-depth invaria…

Let $X$ be a symplectic manifold, let $V\subset X$ be a symplectic crossing divisor carrying the symplectic form, and let $D\subset X\setminus V$. The intrinsic symplectic cochains…

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