Some differential operators on trees and manifolds

The following 3 topics will be presented:

1. Image of the Neumann Laplacian on a tree. We shall discuss how these two operators are related based on a joint work with W.D.Evans (Cardiff, U.K.).

2. Limit of the Neumann Laplacians on shrinking domains. It will be shown that the solution of the limiting equation on ? satisfies the Kirchhoff boundary condition on each vertex of the tree.

3. 2-scale convergence on low-dimensional manifolds in Rn . As an analogy of the previous topic, we shall discuss 2-scale convergence of a function, its derivatives and solutions of differential equations when the domain shrinks to a lower dimensional manifold based by a recent joint work with Willi Jager (Heidelberg, Germany).