We prove that Mark Gross’ [8] topological Calabi-Yau compactifications can be made into symplectic compactifications. To prove this we develop a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. This produces pairs of compact symplectic 6-manifolds homeomorphic to mirror pairs of Calabi-Yau 3-folds together with Lagrangian fibrations whose underlying integral affine structures are dual.

We consider an embedded convex compact ancient solution t to the curve shortening flow in R2. We prove that t is either a family of contracting circles, which is a type I ancient solution, or a family of evolving Angenent ovals, which is of type II.

We study the asymptotic of the Bergman kernel of the spinc Dirac operator on high tensor powers of a line bundle.

In this paper we start the classification of strongly bracket-generated sub-symmetric spaces. We prove a structure theorem and analyze the nilpotent case.

Making use of the extended flux homomorphism defined in [13] on the group Symp§g of symplectomorphisms of a closed ori- ented surface §g of genus g ≥ 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to g and we show that they are highly non- trivial. We also prove that the second homology of the group Ham§g of Hamiltonian symplectomorphisms of §g, equipped with the discrete topology, is very large for all g ≥ 2.