We explore the connection between the analysis of operators with the geometry and the boundary of the space.  The operators in questions are Laplacians, the geometrical input comes via an intrinsic metric of the Laplacian and the boundary typically arises abstractly from the Laplacian via potential theory and harmonic functions. The two directions give rise to the following two parts of the project:
(A)  Higher order Laplacians on simplicial complexes.
(B)  Harmonic functions of Laplacians arising from Dirichlet spaces.