In this talk, the relationship between integrable systems and invariant curve flows is studied. It is shown that many integrable systems including the well-known integrable equations and Camassa-Holm type equations arise from the non-stretching invariant curve flows in Klein geometries. The geometrical formulations to some properties of integrable systems are also given.

The p-adic L-function attached to an elliptic curve with split, multiplicative reduction at the prime p will vanish at s=1. This is an example of what we call a "trivial zero." This talk will outline the way that Glenn Stevens and I proved a formula for the derivative at s=1 for that function.

Optimal Delaunay triangulations (ODTs) are optimal meshes minimizing the inter- polation error to a convex function in Lp norm. We shall present several applications of ODT.
1. Mesh smoothing and optimization. Meshes with high quality are obtained by minimizing the interpolation error in a weighted L1 norm.

2. Anisotropic mesh adaptation. Optimal anisotropic interpolation error esti- mate is obtained by choosing anisotropic functions. The error estimate is used to produce anisotropic mesh adaptation for convection-dominated problems.

3. Sphere covering and convex polytope approximation. Asymptotic exact and sharp estimate of some constant in these two problems are obtained from ODT.

4. Quantization. Optimization algorithms based on ODT are applied to quanti- zation to speed up the processing.