We use the quilt formalism of Mau, Wehrheim, and Woodward to give a sufficient condition for a finite collection of Lagrangian submanifolds to split-generate the Fukaya category, and deduce homological mirror symmetry for the standard $4$-torus. As an application, we study Lagrangian genus $2$ surfaces ${\Sigma }_2\subset T^4$ of Maslov class zero, deriving numerical restrictions on the intersections of ${\Sigma }_2$ with linear Lagrangian $2$-tori in $T^4$

Semicalibrated currents in a Riemannian manifold are currents that are calibrated by a comass-$1$ differential form that is not necessarily closed. This extension of the classical notion of calibrated currents is motivated by important applications in differential geometry such as special Legendrian currents, for example. We prove that semicalibrated integer multiplicity rectifiable $2$-cycles have a unique tangent cone at every point. The proof is based on the introduction of a new technique that might be useful for other first-order elliptic problems

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire map of the form $f(z)=P(z)\exp (Q(z))$, where $P$ and $Q$ are polynomials of arbitrary degrees. (We allow the case $Q=0$.) Building upon a method pioneered by M. Shishikura, we show that if $f$ has a Siegel disk of bounded-type rotation number centered at the origin, then the boundary of this Siegel disk is a quasi circle containing at least one critical point of $f$. This unifies and generalizes several previously known results

We prove that a smooth compact submanifold of codimension $2$ immersed in $R^n,n\ge 3,$ bounds at most finitely many topologically distinct, compact, nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman