We prove that Alexandrov's conjecture relating the area and diameter of a convex surface holds for the surface of a general ellipsoid. This is a direct consequence of a more general result which estimates the deviation from the optimal conjectured bound in terms of the length of  the cut locus of a point on the surface. We also prove that the natural extension of the conjecture to general dimension holds among closed convex spherically symmetric Riemannian manifolds. Our results are based on a new symmetrisation procedure which we believe to be interesting in its own right.

This is joint work with Pedro Freitas accepted for publication in the Tohoku Mathematical Journal, preprint on http://arxiv.org/abs/1406.0811.

We study the holomorphic embedding problem from a compact real algebraic hypersurface into a shpere. By our theorem, for any integer $N$, there is a family of compact real algebraic strongly pseudoconvex hypersurfaces in $C^2$ , none of which can be locally holomorphically embedded into the unit sphere in $C^N$.  This shows that the Whitney (or Remmert) type embedding theorem in differential topology(or in the Stein space theory, respectively) does not hold in the setting above. This is a joint work with Xiaojun Huang and Xiaoshan Li. 

We begin to discuss iteration theory for forcing, with the ultimate goal to study countable support iterations of proper forcing.