10 problems

Let $L$ be a smooth link. Call $L$ orientably exact Lagrangian fillable if it has a Legendrian representative in the standard contact $\mathbb{R}^3$ admitting an orientable exact L…

Casals' conjecture. The standard Legendrian Hopf Link should have exactly two Lagrangian fillings.

Casals' folding conjecture. For Legendrian links of type $\mathsf{A}_{2n-1}$, $\mathsf{D}_{n+1}$, $\mathsf{E}_6$, and $\mathsf{D}_4$, Lagrangian fillings having certain…

Let $\beta$ be a positive braid, let $L_\beta$ be its braid-positive link closure, and let the corresponding Legendrian closure be represented by the front diagram $f_\beta$. A Leg…

Let $\Lambda$ be a Legendrian link with marked points, let a Morse function be chosen, and let $\mathcal{A}ug_+$ and $\mathcal{C}$ be the associated strictly unital proper…

Let $n_1\geq n_2>2$, and consider the $T_{n_1,n_2,2}$ quivers associated with the indicated Legendrian links. Let $\Gamma_\tau$ be the subgroup generated by the tail automorphisms.…

The examples above concern Legendrian links admitting non-orientable Lagrangian fillings, and distinguish only finitely many such fillings in each construction. Finiteness conjectu…

Let $\mathcal{X}$ be a globally hyperbolic spacetime whose Cauchy surface is not homeomorphic to the sphere. Let $x,y\in\mathcal{X}$ be causally related points, and consider future…

Finiteness conjecture. The Legendrian torus link $\Lambda(n,m)$ has finitely many Lagrangian fillings whenever

Let $({X}^{m+1},g)$ be a globally hyperbolic spacetime whose Cauchy surface is diffeomorphic to an open subset of $\mathbb R^m$. When $m=2$, let $x,y\in X$ be events and let…