I present some new dispersive estimates for Schroedinger's equation with a time-dependent potential, together with applications.

For a set of polynomials F, we define their bilinear complexity as the smallest k so that F lies in an ideal generated by k bilinear polynomials. The main open problem is to estimate the bilinear complexity of the single polynomial $\sum_{i,j}x_i^2 y_j^2$. This question is related to the classical sum-of-squares problem as well as to problems in arithmetic circuit complexity. We will focus on related sets of polynomials and prove some lower and upper bounds on their bilinear complexity.

We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field.
Motivated by work of Kato and others for n=3, we show that local-global principles hold for
H^n(F, Z/mZ(n-1)) for all n>1.
In the case n=1, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for H^1(F,G), a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups G.

This is from joint works with D. Knopf and I. M. Sigal. In this talk I will present a new strategy in studying neckpinching of mean curvature flow. Different from previous results, we do not use backward heat kernel, entropy estimates or subsequent convergence, instead we apply almost precise estimates, invented in the past few years, to obtain the first result on asymmetric surface.