We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of $T^3$ similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of $T^3$ are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured $J$-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on $T^{*}{T}^2$ is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result [G] for contact manifolds with positive Giroux torsion

We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary ${\mathbb{Z}}^d$-graded binomial ideal $I$ in $\mathbb{C}[{\partial }_1,\dots ,{\partial }_n]$ along with Euler operators defined by the grading and a parameter $\beta \in {\mathbb{C}}^d$. We determine the parameters $\beta$ for which these $D$-modules (i) are holonomic (equivalently, regular holonomic, when $I$ is standard-graded), (ii) decompose as direct sums indexed by the primary components of $I$, and (iii) have holonomic rank greater than the rank for generic $\beta$. In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arrangement in ${\mathbb{C}}^d$. In the special case of Horn hypergeometric $D$-modules, when $I$ is a lattice-basis ideal, we furthermore compute the generic holonomic rank combinatorially and write down a basis of solutions in terms of associated $A$-hypergeometric functions. This study relies fundamentally on the explicit lattice-point description of the primary components of an arbitrary binomial ideal in characteristic zero, which we derive in our companion article [DMM]

We explore some geometric and analytic consequences of a curvature condition introduced by Ma, Trudinger, and Wang in relation to the smoothness of optimal transport in curved geometry. We discuss a conjecture according to which a strict version of the Ma-Trudinger-Wang condition is sufficient to prove regularity of optimal transport on a Riemannian manifold. We prove this conjecture under a somewhat restrictive additional assumption of nonfocality; at the same time, we establish the striking geometric property that the tangent cut locus is the boundary of a convex set. Partial extensions are presented to the case when there is no pure focalization on the tangent cut locus

We investigate $C^1$-regularity of minimizers to $\int F(\nabla u)\mathrm{dx}$ in two dimensions for certain classes of nonsmooth convex functionals $F$. In particular, our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of Kenyon, Okounkov, and others