The Reeb graph of a continuous function on a manifold is obtained by contracting each level set to a point. It reflects the essential topological information of the manifold and the function defined on the manifold. We proved that for a connected orientable 2-manifold of genus g, the Reeb graph has exactly g loops. For orientable 2-manifolds with h boundary components and for non-orientable 2-manifolds with and without boundary, the number of loops is between 0 and 2g+h-1 and depends on the function. We also gave an algorithm that constructs the Reeb graph in time O(n log n), where n is the number of edges in the triangulation of the 2-manifold.

We are working on extensions of these results to 3-manifolds and to time-varying functions on 2-manifolds. The latter extension is based on so-called Jacobi curves, which are the paths traced out by the critical points of the time-varying function. More generally, the Jacobi set of a collection of k+1 Morse functions is the set of critical points of our function restricted to the intersection of the level sets of the others. These sets are of independent interest and relate to folds studied in singularity theory and to relative equilibria in classical mechanics. We developed a combinatorial algorithm for computing piecewise linear approximations of Jacobi sets and work on meshing methods that improve the approximation of the smooth Jacobi sets.