The isometric embedding problem: manifolds vs. polyhedra, rigidity vs. flexibility

This talk will mostly be a survey on what is known about the isometric embedding problem for various types of metric spaces.  We will compare and contrast the known results for manifolds and polyhedra, and we will classify these results into two types:  those that are "flexible" vs. those that are "rigid".  If time permits, we will discuss what is known for more general types of metric spaces.