We show that the set of C^∞ Riemannian metrics on S² or ℝP² whose geodesic flow has positive topological entropy is open and dense in the C² topology. The proof is partially based on an analogue of Franks' lemma for geodesic flows on surfaces.

We generalize the classical Szpiro inequality to the case of a semistable family of hyperelliptic curves. We show that for a semistable symplectic Lefschetz fibration of hyperelliptic curves of genus g, the number N of nonseparating vanishing cycles and the number D of singular fibers satisfy the inequality N ≤ (4g + 2)D.

If a group acts by uniformly quasi-Möbius homeomorphisms on a compact Ahlfors n-regular space of topological dimension n such that the induced action on the space of distinct triples is cocompact, then the action is quasisymmetrically conjugate to an action on the standard n-sphere by Möbius transformations.

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkähler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the theory of Lagrangian intersection.

We also introduce and study extensively a normalized Legendre transformation of Lagrangian subvarieties under a birational transformation of projective hyperkähler manifolds. We give a Plücker type formula for Lagrangian intersections under this transformation.

Let X and Y be smooth projective varieties over ℂ. They are called D-equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, and K-equivalent if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.