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.

Algebraic complexity theory is the study of the computational difficulty of infinite families of polynomials -- a generalization of the computational complexity of decision problems. In this theory, there are analogues of the usual complexity classes P and NP, as well as different NP-complete problems. The main aim is to find complexity lower bounds for certain specific families, such as the families for matrix multiplication and the permanent family. There are different approaches using algebraic geometry and representation theory to attack such lower bound problems. This talk will be a brief introduction to this area and its central problems.