32 problems

Let $M$ be a Liouville manifold, let $\Lambda\subset M$ be a subset, and let $\mathcal B$ be a choice of brane data. Write $\lag^{\mathcal B}_\Lambda(M)$ for the associated Lagrang…

Stabilized Lagrangian isotopy conjecture. If $L_1$ and $L_2$ are smoothly ribbon isotopic, then after stabilizing both cobordisms, they become Lagrangian isotopic.

Let $\mathcal K$ denote the symplectic bielliptic surface, and let $\operatorname{Cob}^{trop}(\mathcal K)$ be the subgroup of the Lagrangian cobordism group…

Functoriality conjecture. There is an $L_\infty$ morphism from the $L_\infty$ algebra $((\mathcal{A}^+)^{\operatorname{comm}},\{\ell_k^+\})$ of $\Lambda^+$ to the $L_\infty$ algebr…

Let $L_0$ and $L_1$ be Lagrangian isotopic submanifolds, and let $K:L_1\rightsquigarrow L_0$ be a Lagrangian cobordism. Let $d(L_0,L_1)$ denote the minimal flux distance, defined b…

Let $L$ be an exact Lagrangian cobordism in $Y\times\mathbb{R}$ from Legendrians $\Lambda_-$ to $\Lambda_+$, and let $\widetilde{L}$ be its lift to a Legendrian cobordism in…

Weinstein isomorphism conjecture. For Weinstein manifolds, all of these maps should be isomorphisms.

Let $(K,t^-,t^+):(L^+,H^+)\to(L^-,H^-)$ be a Lagrangian cobordism with double bottlenecks. Bottlenecked Floer cohomology conjecture. There exists a version of Floer cohomology…

Let $K:L^+\rightsquigarrow L^-$ be a Lagrangian cobordism, possibly immersed and non-monotone, and let ${CF^\bullet}(L^+,L')$ and ${CF^\bullet}(L^-,L')$ denote the relevant Floer c…

Let $K$ be a Lagrangian cobordism, and consider the Floer-theoretic maps associated to its exact-homotopy setting. Shadow-bound conjecture. The decrease in valuation is bounded by…

Let $\Lambda_{-}$ and $\Lambda_{+}$ be the Legendrian knots described in the satellite construction, and let $P\subset J^1S^1$ be the Legendrian clasp tangle used to form their sat…

Cobordism description conjecture. This Lagrangian cobordism construction describes the Grothendieck group

Duality compatibility conjecture. For $j=1,\ldots,s-1$, the quasi-isomorphisms

Biran–Cornea identification conjecture. The categories $T^s$ of Biran and Cornea are the discrete colored planar operads associated to the $s$-dot constructions of…

Paracyclic compatibility conjecture. The map is compatible with the paracyclic action on the $s$-dot construction.

Compatibility conjecture. The filtration of the vertical-collar theorem defines a map of colored planar $\infty$-operads to the colored planar $\infty$-operad obtained from the $s$…

Given a Legendrian knot $\Lambda$ in a contact $3$-manifold $(Y,\xi)$, with a decoration $P$ consisting of two points, its LOSS invariant is … where $\mathfrak{t}_\xi$ is the…

Biran–Cornea's conjecture. Every exact Lagrangian cobordism is Hamiltonian isotopic to a Lagrangian suspension.

Let $\LCob_q$ be the category of Lagrangian cobordisms and let $\tsA$ be the corresponding target category of Jacobi-diagram-valued objects. The LMO functor is … Its construction u…

Let $(W;L_0,L_1)$ be an exact Lagrangian cobordism, and let a Lagrangian suspension denote the suspension associated to a Hamiltonian isotopy. Biran and Cornea's conjecture. Any ex…

Let the Legendrian invariants defined by Ozsváth, Szabó and Thurston and by Lisca, Ozsváth, Stipsicz and Szabó be the GRID and LOSS Legendrian invariants, respectively. A Lagrangia…

Equivalence conjecture. The study of generating family-compatible Lagrangian cobordisms between Legendrian submanifolds should be tantamount to the study of zero-Maslov Lagrangian…

Nadler–Tanaka's wrapped-Fukaya conjecture. The continuation-map functor induces an equivalence of stable $\infty$-categories

Nadler–Tanaka's cosheaf conjecture. The functor

Nadler–Tanaka's microlocal sheaf conjecture. There is a canonical equivalence of stable $\infty$-categories