The error of piecewise polynomial interpolation on anisotropic meshes
is determined by the geometric features of the elements (size,
orientation,
and aspect ratio) and the higher order derivatives of the interpolated
function. In this talk we introduce some quantities to measure the
anisotropic behavior of higher order derivative tensors.
Based on these measures, we derive the error estimates for interpolations
on anisotropic meshes that are quasi-uniform under a given Riemannian
metric.
By using the inertia properties for matrix eigenvalues, we can
identify the optimal mesh metrics leading to the smallest interpolation
error in various norms. Numerical results indicate that the smallest
error is attained exactly on meshes generated with the optimal metrics
as predicted.

String theory has helped to formulate two major new insights in the study of singular algebraic varieties. The first -- which also arose from symplectic geometry -- is that families of Kaehler metrics are an important tool in uncovering the structure of singular algebraic varieties. The second, more recent insight -- related to independent work in the representation theory of associative algebras -- is that one's understanding of a singular (affine) algebraic variety is enhanced if one can find a non- commutative ring whose center is the coordinate ring of the variety. We will describe both of these insights, and explain how they are related to string theory.